GIFT  OF 
Prof.   ±i.   P.    Lev/is 


A  MANUAL  OF  PHYSICS 


BEING 


AN  INTRODUCTION  TO  THE  STUDY  OF 
PHYSICAL  SCIENCE. 


for  the  ibe  af  JSttitoeraitj)  <Stufceut0. 


BY 

WILLIAM  PEDDIE,  D.Sc.,  F.R.S.E., 

M 

\SSISTANT   TO   THE    PROFESSOR    OF   NATURAL   PHILOSOPHY    IN   THE    UNIVERSITY    OF 
EDINBURGH. 


NEW  YORK  :    G.  P.  PUTNAM'S  SONS. 

LONDON:   BAILLIERE,   TINDALL  &  COX. 

1892. 


£> 


P  E  E  P  A  C  E . 


THE  best  advice  which  can  be  given  to  a  student  of  physics  regard- 
ing the  books  which  he  should  read  is  to  use  separate  works 
written  on  the  various  branches  of  the  subject  by  the  leading 
physicists  of  the  day.  Yet  this  advice  has  one  evident  disadvantage. 
The  student  who  follows  it  may  not  get  so  complete  a  view  of  the 
essential  unity  and  interdependence  of  the  various  branches  of  his 
subject  as  it  is  desirable  that  he  should.  And,  besides  this,  there  is 
no  doubt  that  a  small  volume  which  gives,  as  far  as  is  possible,  a 
review  of  the  elements  of  the  whole  subject,  is  a  desideratum  to  the 
student  while  in  attendance  on  University  classes.  I  have  under- 
taken the  writing  of  this  work  in  the  hope  that  it  may  to  some 
extent  meet  that  want. 

I  have  throughout  endeavoured  to  bring  into  prominence  the 
necessity  for,  and  the  value  of,  scientific  hypotheses — a  matter 
regarding  which  very  hazy  notions  are  only  too  common. 
-  It  has  also  been  my  aim  to  make  the  treatment  of  the  mathe- 
matical portions  of  the  text  as  simple  as  possible.  In  this  connection 
I  have  not  adopted  the  process  which  has  recently  been  termed 
'  calculus-dodging,'  for  the  reason  that  the  elementary  methods  of 
the  calculus  are  more  simple,  certainly  are  more  natural,  than  the 
methods  by  which  they  are  usually  supplanted. 

At  the  same  time  it  may  be  well  to  remark  that  any  student, 
who  desires  to  do  so,  may  simply  assume  the  results  of  the  mathe- 
matical portions,  and  use  the  remainder  (which  is  much  the  larger 
part)  of  the  text  in  his  study  of  experimental  physics. 

In  writing  a  text-book  on  general  physics  it  is  impossible,  if 
justice  is  to  be  done  to  the  subject,  to  avoid  borrowing  methods 
from  the  writings  of  the  masters.  In  this  respect  I  have  to  acknow- 
ledge my  indebtedness  to  the  works  of  v.  Helmholtz,  Clerk- 


IV  PREFACE. 

Maxwell,  Thomson,  Tait,  and  others.  This  is  perhaps  most  evident 
in  Chapter  XXVI.,  where  I  have  made  use  of  the  very  simple  form 
in  which  Tait  has  presented  the  analytical  treatment  of  the  theory 
of  Thermodynamics ;  and  in  Chapter  XI.,  where  I  have  adopted 
his  mode  of  discussing  the  compressibility  and  rigidity  of  solids. 
This  apart,  I  have  endeavoured,  whether  successfully  or  not,  to 
present  the  various  subjects  in  as  fresh  a  manner  as  I  could. 

My  indebtedness  to  Professor  Tait  has  also  been  very  great  in  the 
matter  of  criticism,  which  he  kindly  afforded  me  on  various  points 
while  the  book  was  passing  through  the  press.  I  have  also  to 
acknowledge  with  thanks  the  kindness  of  Mr.  J.  B.  Clark,  M.A., 
F.B.S.E.,  Physical  Master  in  Heriot's  Hospital,  Edinburgh,  in 
reading  the  work  both  in  manuscript  and  in  proof.  His  careful 
revision  has  resulted  in  the  elimination  of  a  number  of  defects  which 
had  escaped  my  notice. 

WILLIAM  PEDDIE 

EDINBUKGH  UNIVERSITY, 
November,  1891. 


CONTENTS. 


CHAPTER  I. 
INTRODUCTORY:  THE  PHYSICAL  UNIVERSE. 

PAGE 

Physical  science. — Test  of  reality  :  conservation. — Matter  :  its 
conservation  :  its  inertia. — Energy  :  its  transformation  : 
its  conservation  :  its  degradation  .  .  .  1 — 8 

CHAPTER  II. 

THE    METHODS    OF    PHYSICAL    SCIENCE. 

Observation  and  experiment. — Cause  and  effect. — Hypothesis 
and  theory. — Crucial  test. — Argument  from  analogy — 
Principle  of  stable  equilibrium. — Errors  of  observation. — 
Instrumental  errors. — Empirical  laws.  —  The  graphical 
method  .......  9—17 

CHAPTER  III. 

THE    THEORY    OF    CONTOURS,    AND    ITS    PHYSICAL   APPLICATIONS. 

Dimensions. — Contour  points  of  plane  and  tortuous  curves. — 
Examples  of  their  applications  in  physics.  —  Contour 
lines. — Special  physical  applications. — Use  of  trilinear 
co-ordinates  .  .  .  .  .  .  18 — 32 

CHAPTER  IV. 

VARYING   QUANTITIES. 

Dimensions  of  physical  quantities. — Determination  of  rates  of 

variation. — The  inverse  problem  .  .  .     38 — 47 


VI  CONTENTS. 

CHAPTER  V. 

MOTION. 

PAGE 

Position. — Displacement. — Speed  and  velocity. — Acceleration 
of  speed. — Curvature. — Acceleration  of  velocity. — Accelera- 
tion in,  and  perpendicular  to,  the  direction  of  motion. — 
Average  speed  and  velocity. — Resolution  and  composition 
of  velocities  and  accelerations. — Motion  of  projectiles. — 
The  hodograph. — Moments. — Acceleration  perpendicular 
to  the  radius-vector. — Simple  harmonic  motion. — Composi- 
tion of  simple  harmonic  motions. — Wave  motion  along  a 
line.  —  Rotation. — Alteration  of  co-ordinates  because  of 
rotation. — Uniplanar  motion  of  a  rigid  body. — Motion  of  a 
rigid  body  in  space. — Composition  of  angular  velocities. — 
Displacement  of  the  parts  of  a  non-rigid  body. — Strain.— - 
Homogeneous  strain. — Shear. — Non-homogeneous  strain. — 
Motion  of  fluids. — Fluid  circulation. — Vortex  motion  48 — 78 

CHAPTER  VI. 

MATTER   IN    MOTION. 

Force. — The  laws  of  motion. — The  first  and  second  laws. — 
Examples.  —  Dynamical  similarity.  —  The  third  law. — 
Centre  of  inertia. — Moment  of  a  force  and  of  inertia. 
— Rotational  equilibrium. — Propagation  of  motion  through 
a  non-rigid  solid. — Motion  of  a  perfect  fluid. — Equilibrium 
of  a  fluid. — Propagation  of  surface-waves  in  liquid  .  79 — 100 

CHAPTER  VII. 

PROPERTIES    OF   MATTER. 

Definitions  of  matter. — States  of  matter. — General  properties. 
— Special  properties. — Specific  properties. — Molecules  and 
atoms. — Molecular  forces  ....  101 — 106 

CHAPTER  VIII. 

GRAVITATION. 

Weight  and  mass. — Kepler's  laws  with  Newton's  deductions. — 
The  law  of  gravitation. — Newton's  theorems  regarding 
spherical  shells. — Mean  density  of  the  earth. — Hypotheses 


CONTENTS.  Vll 

PAGE 

framed  to  explain  gravitation. — The  nebular  hypothesis. — 
Potential. — ' Gravitational  potential.  —  Equipotential  sur- 
faces.— Lines  and  tubes  of  force. — Special  theorems  107 — 124 

CHAPTEE  IX. 

PROPERTIES    OF    GASES. 

Compressibility. — Boyle's  law. — Compressibility  of  a  perfect 
gas. — Deviations,  from  Boyle's  law.  —  Compression  of 
vapours.  —  Elasticity.  —  Viscosity.  —  Diffusion.  —  Effusion. 
—Transpiration  .....  325—133 

CHAPTEE  X. 

PROPERTIES    OF   LIQUIDS. 

Compressibility. — Elasticity. — Viscosity  and  viscidity. — Diffu- 
sion. —  Osmose.  —  Dialysis.  —  Cohesion.  —  Capillarity.  — 
Surface-tension  .....  134—144 

CHAPTEE  XI. 

PROPERTIES    OF    SOLIDS. 

Compressibility      and      rigidity.  —  Elasticity.  —  Viscosity.  — 

Cohesion — Tenacity    .  .  .  .  .     145 — 153 

CHAPTEE  XII. 

THE    CONSTITUTION    OF   MATTER. 

The  hard  atoms  of  Lucretius. — Boscovich's  centres  of  force. — 
Vortex  atoms. — Heterogeneity  of  matter. — Structure  of 
crystals. — Molecular  nature  of  matter. — Size  of  molecules 
and  the  range  of  molecular  forces  .  .  .  154 — 162 

CHAPTEE  XIII. 

THE    KINETIC   THEORY   OF   MATTER. 

Kinetic  theory  of  gases. — Gaseous  pressure. — Boyle's  law. — 
Avogadro's  and  Charles'  laws. — Diffusion. — Thermal  con- 
ductivity.— Viscosity. — Evaporation. — Dissociation,  etc. — 
Difficulties  of  the  theory. — General  kinetic  theory  of 
matter  163—170 


Vlll  CONTENTS. 

CHAPTEE  XIV. 

SOUND. 

PAGE 

Propagation.  —  Intensity.  —  Eeflection.  —  Kefraction. —  Diffrac- 
tion. —  Interference.  —  Pitch.  —  Musical  intervals. — Vibra- 
tions of  rods. — Vibrations  of  plates. — Vibrations  of  strings. 
—Vibrations  of  air-columns. — Speed  of  sound  in  a  gas — 
Partial  tones.  —  Resonance.  —  Quality.  —  Beats.  —  Con- 
sonance and  dissonance. — Combination  tones. — Dissonance 
of  pure  tones  ....  .  171 — 191 

CHAPTEE  XV. 

LIGHT  :    INTENSITY,    SPEED,    THEORIES. 

Eectilinear    propagation.  —  Intensity.  —  Speed.  —  Theories.  — 

Colour  ......     192—197 

CHAPTEE  XVI. 

LIGHT  :     REFLECTION,    REFRACTION,    DISPERSION. 

Laws  of  reflection. — Eeflection  from  plane  surfaces. — Eeflection 
from  curved  surfaces. — Caustics. — Focal  lines. — Theoretical 
deductions  of  the  laws  of  reflection. — Laws  of  refraction. 
— Eefraction  through  a  plane  surface. — Focal  lines. — 
Caustics. — Eefraction  through  parallel  layers. — Mirage. — 
Prisms. — Eefraction  through  spherical  surfaces. — Lenses. 
Oblique  refraction. — Formation  of  images  by  lenses. — 
Dispersion. — Aberration.  — Achromatism.  —  Eainbows.  — 
Halos. — Theoretical  deductions  of  the  laws  of  refraction. 
— Hamilton's  characteristic  function  .  .  198 — 227 

CHAPTEE  XVII. 

RADIATION    AND    ABSORPTION  :     SPECTRUM   ANALYSIS. — ANOMALOUS 
DISPERSION. — FLUORESCENCE. 

Equality  of  emissivity  and  absorptive  power.  —  Spectrum 
analysis. — Law  of  absorption. — Body  colour. — Dichroism. 
— Surface  colour. — Metallic  reflection. — Anomalous  dis- 
persion. —  Fluorescence.  —  Phosphorescence.  —  Theories  of 
dispersion  ......  228—240 


CONTENTS.  IX 

CHAPTEE  XVIII. 

INTERFERENCE. — DIFFRACTION. 

PAGE 

Principle  of  interference. — Young's  experiment. — Fresnel's 
experiment. — Lloyd's  experiment. — Fresnel's  biprism. — 
Coloured  interference  bands. — Displacement  of  bands  by 
refracting  media. — Interference  bands  in  spectra. — Colours 
of  thin  plates. — Newton's  rings. — Colours  of  mixed  plates. 
— Colours  of  thick  plates. — Diffraction. — Effect  of  a  recti- 
linear wave.  —  Effect  of  plane  and  spherical  waves. — 
Diffraction  at  a  straight  edge. — Diffraction  at  a  narrow 
slit. — Diffraction  at  a  circular  aperture. — Zone  plates. — 
Diffraction  at  an  opaque  disc.  —  Coronae.  —  Young's 
Eriometer. — Diffraction  gratings  .  .  .  241 — 261 

CHAPTEE  XIX. 

DOUBLE    REFRACTION. — POLARISATION. 

Double  refraction. — Huyghens'  construction. — Special  sections 
of  the  surface. — Polarisation. — Laws  of  polarisation  by 
reflection  and  refraction. — Direction  of  vibration  in 
polarised  light. — Eeflection  and  refraction  of  polarised 
light. — Plane,  circular,  and  elliptic  polarisation. — Nature 
of  common  light. — Metallic  reflection. — Double  Tefraction 
in  biaxal  crystals. — Fresnel's  theory  of  double  refraction. 
—  Conical  refraction.  —  Interference  of  polarised  light. 
— Colours  of  crystalline  plates. — Haidinger's  brushes. — 
Artificial  production  of  the  doubly  refracting  structure. — 
Eotatory  polarisation. — Polarising  prisms  .  .  265 — 295 

CHAPTEE  XX. 

THE   NATURE   OF   HEAT. 

Eadiant  heat.  —  Its  identity  with  light.  —  Heat  in  material 

bodies. — Hypothesis  of  molecular  vortices    .  .     296 — 300 

CHAPTEE  XXI. 

RADIATION   AND   ABSORPTION    OF   HEAT. 

Prevost's  theory  of  exchanges.  —  Stewart's  and  Kirchhoff's 
extension  of  Prevost's  theory. — Laws  of  radiation  of  heat 
at  constant  temperature.  —  Heat  spectra.  —  Eadiation  at 
different  temperatures. — Solar  radiation. — Eadiation  from 
moving  bodies  ...  .  301 — 309 


CONTENTS. 


CHAPTER  XXII,. 

EFFECTS    OF    THE    ABSORPTION    OF    HEAT  :     DILATATION    AND    ITS 
PRACTICAL    APPLICATIONS. 

PAGE 

Temperature. — Dilatation  of  solids. — Dilatation  of  liquids. — 
Dilatation  of  gases. — Absolute  zero  of  temperature. — 
Measurement  of  temperature  .  .  .  310 — 323 


CHAPTER  XXIII. 

EFFECTS  OF  THE  ABSORPTION  OF  HEAT  :  CHANGE  OF  TEMPERATURE 
AND  CHANGE  OF  STATE. 

Unit  of  heat. — Specific  heat. — Thermal  capacity. — Specific 
heat  of  solids  and  liquids. — Law  of  Dulong  and  Petit. — 
Specific  heat  of  gases  and  vapours. — Change  of  molecular 
state. — Latent  heat. — Fusion  and  solidification. — Latent 
heat  of  fusion. — Evaporation  and  condensation. — Latent 
heat  of  vaporisation. — Formation  of  dew. — Continuity  of 
the  liquid  and  gaseous  states. — Critical  temperature. — 
Solution. — Freezing  mixtures. — Dissociation  and  chemical 
combination  ......  324 — 343 

CHAPTER  XXIV. 

CONDUCTION  AND  CONVECTION  OF  HEAT. 

Conduction. — Conductivity. — Measurement  of  conductivity. — 
Conduction  through  the  earth's  crust. — Conduction  in 
crystalline  bodies. — Conduction  in  liquids  and  gases. — 
Convection  ......  344—354 

CHAPTER  XXV. 
THERMODYNAMICS:   HEAT  AND  WORK. 

Mechanical  equivalent  of  heat. — First  law  of  thermodynamics. 
— Carnot's  complete  cycle  of  operations. — Carnot's  rever- 
sible cycle. — Reversibility  the  test  of  perfection". — Second 
law  of  thermodynamics. — Absolute  temperature.  —  The 
indicator  diagram  and  its  applications. — Entropy. — 
Total,  available,  and  dissipated  energy. — Thermodynamic 
motivity  ......  355—369 


CONTENTS. 


CHAPTER  XXVI. 

PAGE 
THERMODYNAMICAL    RELATIONS  370 — 376 


CHAPTER  XXVII. 

ELECTROSTATICS. 

Electrification  by  friction. — Conductors  and  non-conductors.^- 
Fundamental  phenomena  presented  by  electrified  bodies. 
— Positive  and  negative  electricity. — The  gold-leaf  electro- 
scope.— Electrification  by  contact  and  by  induction. — 
Electric  quantity. — Continued  production  of  electricity. — 
The  electrophorus. — Law  of  electric  attraction  and  repul- 
sion.— Electric  potential. — Electromotive  force. — Capacity. 
-  — Condensers. — Specific  inductive  capacity. — Distribution 
of  electricity  on  conductors. — Electric  density. — Electric 
images. — Lines  of  electric  force. — Electric  induction. — 
Tubes  of  induction. — Electric  energy. — Electric  absorption. 
— Disruptive  discharge. — Atmospheric  electricity. — Pyro- 
electricity. — Electrification  by  contact. — The  electrometer. 
— Electric  machines  .....  377 — 412 

CHAPTER  XXVIII. 

THERMO-ELECTRICITY. 

Thermo-electric  phenomena. — Laws  of  thermo-electric  circuits. 
— Variation  of  the  electromotive  force  with  temperature. 
— The  thermo-electric  diagram. — The  Peltier  effect. — 
The  Thomson  effect. — Further  discussion  of  the  thermo- 
electric diagram  .....  413 — 423 

CHAPTER  XXIX. 

ELECTRIC     CURRENTS. 

Convection  current  between  charged  conductors. — Flow  of  elec- 
tricity in  metallic  conductors. — Ohm's  law. — Kirchhoff's 
laws. — Electrolytic  conduction. — Faraday's  laws. — Polari- 
sation.— Ohm's  law  in  electrolytes. — Production  of  electric 
currents. — Primary  cells. — Secondary  cells.— Transforma- 
tions of  electric  energy  in  conducting  circuits. — Joule's 
law.  —  Measurement  of  electromotive  force,  current 
strength,  and  resistance  ....  424 — 440 


Xil  CONTENTS. 

CHAPTEE  XXX. 

MAGNETISM. 

PAGE 

Fundamental  phenomena. — North  and  south  magnetism. — 
Paramagnetic  and  diamagnetic  bodies. —  Magnetism  a 
molecular  phenomenon. — The  law  of  magnetic  attraction 
and  repulsion. — Poles,  axis,  and  magnetic  moment  of  a 
magnet. — Lines  of  magnetic  force. — Magnetic  potential. — 
Magnetic  intensity. — Magnetic  induction. — Permeability 
and  susceptibility. — Residual  magnetism. — Reteiitiveness. 
— Coercive  force. — Eelation  connecting  magnetisation  and 
magnetising  force. — Hysteresis. — Effects  of  vibration  and 
of  temperature.  —  Effects  of  stress.  —  Magnetometric 
measurements.  —  Terrestrial  magnetism.  —  Theories  of 
magnetism  .  .  .  .  .  .  441 — 462 

CHAPTER  XXXI. 

ELECTROMAGNETISM,    ETC. 

Oersted's  discovery. — Magnetic  action  of  closed  electric  circuits. 
— Electrodynamic  action  on  an  electric  circuit. — Linear 
circuits. — Circular  circuits. — Solenoids. — Ampere's  hypo- 
thesis regarding  magnetism. — Continuous  rotation  under 
electromagnetic  force. — Electric  motors. — Electromagnetic 
induction. — The  dynamo. — Electrokinetic  energy. — The 
galvanometer. — The  ballistic  method. — Electric  and  mag- 
netic units  ......  463—480 

CHAPTER  XXXII. 

ELECTROMAGNETIC    THEORY    OF   LIGHT. 

Magnetic  rotation  of  the  plane  of  polarisation  of  light. — 
Hypothesis  of  molecular  vortices. — Hall's  effects. — Kerr's 
effects. — Electromagnetic  theory  of  light. — Electromagnetic 
waves.  ...  .  481—489 

CHAPTER  XXXIII. 

THE    ETHER. 

Direct  action  at  a  distance. — Action  through  a  medium. — 
Rigidity  and  density  of  the  ether. — Elastic-solid  theory. — 
Contractile  ether.— Electromagnetic  ether.— Dilatancy  421 — 496 

INDEX  497—501 


E  E  E  A  T  A. 

Page  17,  line  11,  for  '  wrong  '  redd  '  discordant.' 
,,     26,    „      9,  for   '  the  closed  curve  at  '  read  '  a  closed  curve 

surrounding.' 

,,     28,    ,,      8  from  foot,  for  '  here  '  read  l  there.' 
,,     49,    ,,      1,  for  *  relative  position  of  P  '  read  '  position  of  P 

relative.' 


149,  Equation  (2),/or  ~0  read  - 

173,  ,,         (4),  for  *  pd  '  read  *  rf/o.' 

183,  line  12,  for  '  63  '  read.'  64.' 

198,  ,,  13,  /or  '  refraction  '  read  '  reflection.' 

207,  „  I,  for  'p>  read  'If  p.' 

248,  „  5,  for  '  Fox,'  read  '  Fox-'. 

257,  „  3,  for  *  e.2,  «?2  '  read  *  <?i,  e2-' 

273,  „  15,  /or  'XXXIII.'  read  'XXXII.' 

395,    „    12,  /or-  read  -. 


A  MANUAL   OF  PHYSICS. 


CHAPTEE  I. 

INTRODUCTORY  :    THE    PHYSICAL    UNIVERSE. 

1.  ALL  processes  which  occur  in  the  universe  may  be  classified  as 
physical  or  as  non-physical.  They  appertain  essentially  on  the 
one  hand  to  dead  matter,  or,  on  the  other  hand,  to  matter  which 
possesses  (or  is  possessed  by)  life.  This  statement,  of  course,  is .  a 
mere  definition  of  what  is  meant  by  the  word  physical,  and  is  not 
to  be  regarded  as  being  in  any  sense  the  expression  of  an  opinion 
regarding  the  nature  of  life.  If,  however,  in  the  phenomena  as- 
sociated with  living  bodies  we  meet  with  processes  akin  to  others 
which  occur  in  the  inorganic  world,  we  regard  them  as  being  purely 
physical ;  while  if,  in  these  phenomena,  we  meet  with  processes 
which  are  totally  dissimilar  to  any  with  which  we  are  acquainted 
in  the  phenomena  of  dead  matter,  we  leave  their  further  study  to 
the  biologist. 

2.  It  is  evident,  therefore,  that  the  domain  of  the  physical 
sciences  is  of  immense  extent — of  such  extent  that  no  one,  in  the 
course  of  the  longest  life,  could  hope  to  master  all  its  known  details. 
At  one  time,  indeed  (and  that  not  very  remote),  the  scientist  might 
have  said  after  the  fashion  of  Francis  Bacon  'I  have  taken  all 
knowledge  for  my  realm,'  but  such  a  claim  would  be  impossible 
now.  All  questions  regarding  the  combinations  and  interactions 
of  the  various  kinds  of  matter  are  purely  physical  in  their  nature, 
but  their  study  is  now  left  to  specialists  in  the  department  of 
chemistry.  The  investigation  and  prediction  of  the  .motions  tif 
the  heavenly  bodies  and  the  determination  of  their  physical  constitu- 
tion are  left  to  the  astronomer,  while  the  configuration  of  the  earth 
is  studied  by  the  geographer.  So  also  the  sciences  of  navigation 
and  ship-designing,  of  engineering,  mineralogy,  geology  (in  large 

1 


2  A;  MANUAL    OF    PHYSICS. 

part  at  least);  metieqrology,  and  so  on,  are  purely  physical  sciences, 
the  study  of  which  is  undertaken  by  specialists,  for  all  of  them 
could  never  be  fully  studied  by  any  one  man.  Thus  with  increase 
of  knowledge  the  domain  of  the  physical  sciences  has  been  sub- 
divided, and  the  terms  Physical  Science,  Natural  Philosophy, 
or  Physics,  have  become  restricted  in  meaning,  so  that  they  refer 
merely  to  the  pure  scientific  groundwork  which  underlies  all  the 
more  practical  or  more  highly  specialized  physical  sciences. 

3.  In  commencing  the  study  of  physics  we  have  no  concern  with 
purely  metaphysical  questions  regarding  the   objective  reality  or 
non-reality  of  the  universe.      We  simply  assume  that  it  has  an 
existence  quite  independently  of  the  existence  of  an  observing  mind, 
and   then  proceed   to   examine   the   facts   and  phenomena   which 
make   up  its   entirety.      But,  before   entering  upon   any  detailed 
examination,  it  is  well  that  we  should  take  a  glance  at  our  subject 
as  a  whole  in  order  to  learn  something  of  its  scope  and  of  the 
mutual  relations  of  its  various  parts :  and  this  just  for  the  same 
reason  as  that  which  makes  it  desirable  for  the  traveller   in   an 
Unknown   country  to   examine   it   first   from    the    vantage-ground 
of    some  commanding  height,  so  that  he  may  carry  with  him  in 
his   future   wanderings  therein  a  clear  mental  picture  of   its  dis- 
position. 

4.  A  most  obvious  difficulty  meets  us  at  the  very  outset.     How 
are  we  to  distinguish  between  that  which  has  true  existence  and 
that  which  has  only  the  appearance  of  it — between  the  true  land- 
scape, as  it  were,  and  the  mirage  ?      The   mirage  seems  to  the 
observer  to  be  as  real  as  the  reality.     What,  then,  is  to  be  the  test 
of  true  existence  ? 

In  addition  to  the  assumption  of  the  true  existence  of  the  physical 
universe,  it  is  assumed  that  in  this  universe  there  is  nothing 
which  does  not  occur  according  to  law.  But  this  assumption  is 
not  left  without  support.  The  mere  possibility  of  the  existence  of 
the  so-called  exact  sciences  may  be  taken  as  evidence  of  its  truth- 
Keeping  this  idea  in  view,  we  see  that  no  real  thing  can  appear  in, 
or  disappear  from,  the  universe  in  an  arbitrary  manner.  There- 
fore we  cannot  regard  anything  as  a  reality  unless  we  can 
prove  it  to  be  constant  in  amount,  that  is  (to  use  the  ordinary 
scientific  expression),  unless  we  can  show  that  it  possesses  the 
property  of  conservation.  CONSERVATION  is  our  great  test  of 
reality. 

This  test  being  applied,  it  is  found  that  there  are  two,  and  only 
two,  classes  of  things  in  the  physical  world — MATTER  and  ENERGY — • 
which  sustain  it. 


INTRODUCTORY:  THE  PHYSICAL  UNIVERSE.  3 

5.  Everyone  knows  what  is  meant  by  the  term  the  '  matter  '  or 
'  material '  of  which  a  body  is  composed,  so,  in  the  meantime,  no 
attempt  at  a  definition  is  necessary. 

At  first  sight  it  might  appear  that  matter  is  certainly  not  con- 
served. If  we  weigh  a  lump  of  pure  limestone,  so  as  to  determine 
the  amount  of  matter  in  it,  and  then  heat  it  sufficiently,  we  find  that 
its  weight  has  become  less  during  the  process.  It  would  seem  that 
matter  has  been  lost.  But  what  has  actually  occurred  is  the  decom- 
position of  the  carbonate  of  lime,  by  the  application  of  the  heat,  into 
lime  and  carbonic  acid  gas.  The  latter,  being  colourless,  passes 
off  unnoticed,  and  the  second  weighing  gives  only  the  weight  of 
the  lime.  By  proper  means  the  weight  of  the  gas  may  be  deter- 
mined, and  it  is  then  found  that  the  original  weight  of  the  limestone 
is  equal  to  the  sum  of  the  weights  of  its  constituents.  No  matter 
has  been  lost  in  the  process.  And  in  all  chemical  processes, 
however  complex,  the  same  result  holds  :  indeed,  the  science  of 
chemistry  is  a  possibility  only  in  virtue  of  the  strict  conservation 
of  matter.  Therefore  we  say  that  matter  is  a  real  thing. 

There  are  many  kinds  of  matter  which  differ  from  one  another  to 
a  greater  or  less  extent  in  their  various  physical  properties.  These 
properties  will  be  considered  in  detail  in  subsequent  chapters,  and 
their  variations  from  one  to  another  of  a  few  of  the  most  important 
or  most  peculiar  substances  will  be  indicated  ;  but  the  enumeration, 
classification,  and  investigation  of  the  various  substances  in  nature 
belong  more  to  the  science  of  chemistry  than  to  that  of  physics. 
One  special  substance,  which  pervades  all  others  and  extends 
throughout  the  whole  of  the  visible  universe,  possesses  such  extra- 
ordinary properties,  and  is  of  such  immense  importance  physically, 
that  it  must  receive  separate  treatment.  (Chap.  XXXIII.) 

6.  In  addition  to  the  property  of  conservation,  matter  is  charac- 
terised by  passivity  or  inertness.     It  is  said  to  possess  INERTIA. 
In  other  words,  a  material  body  can  do  nothing  of  itself.     If  at 
rest,  it  cannot  move  unless  something  outside  of  itself  sets  it  in 
motion.     If  moving,  it  cannot  come  to  rest  or  alter  its  motion  in 
any  way  unless  something  external  to  it  produce  that  effect.     This 
property  and  its   consequences  will  be  discussed  under  Newton's 
'  First  Law  of  Motion '  (Chap.  VI.).     It  is  the  distinguishing  charac- 
teristic of  matter.     One  kind  of  matter  can  unite  with  another 
kind  so  as  to  produce  a  compound,  differing  entirely,  it  may  be,  in 
its  properties  from  both  of  its  constituents :   but,  without   some- 
thing external  to  the  matter,  no  such  combination  or  change  could 
occur. 

7.  There  is  every  reason  to  believe  that  the  inertia  of  matter 

1—2 


4  A   MANUAL    OF    PHYSICS. 

cannot  be  overcome,  that  is,  that  the  motion  of  matter  cannot  be 
altered,  unless  motion  is  imparted  to  it  from  some  other  portion 
of  matter,  whether  by  direct  collision  or  otherwise.  It  is  commonly 
said,  in  such  a  case,  that  the  second  body  '  does  work '  upon  the 
first,  or  that  the  first  has  '  work  done  '  upon  it  by  the  second. 
Thus  '  the  doing  of  work '  involves  essentially  the  production  of 
motion.  We  might  quite  consistently  assert  that  the  second  body 
possesses  '  work,'  and  that  it  imparts  '  work  '  to  the  first— the 
transference  of  work  being  that  process  which,  in  ordinary  language, 
is  called  the  performance  of  work.  But  it  is  usual  and  convenient 
to  adopt  the  term  ENERGY  instead ;  for,  although  in  all  likelihood 
the  possession  of  work  or  energy  necessarily  involves  motion  of 
the  material  system  which  possesses  it,  the  moving  system  may 
not  be  evident  to  our  senses,  in  which  case  it  is  convenient  to 
speak  of  the  energy  as  existing  potentially  in  some  connected 
system  which  may  be  at  rest  at  the  time,  but  which  can  be  set 
in  motion  by  having  energy  imparted  to  it  from  the  other.  (See 
next  section.) 

Of  two  bodies  moving  with  a  common  speed,  that  one  with  least 
mass  (quantity  of  matter)  can  do  least  work,  and  the  same  is  true 
of  the  slower  moving  of  two  bodies  which  have  the  same  mass. 
The  work  tends  to  vanish  either  as  the  mass  or  the  speed  becomes 
indefinitely  small ;  and,  so  far  as  experiment  (which  is  the  only  per- 
missible test)  shows,  it  does  not  depend  upon  anything  else  than  the 
mass  and  the  speed. 

Each  unit  of  mass,  moving  with  a  given  speed,  is  found  to  possess 
the  same  amount  of  energy  ;  and,  therefore,  the  energy  of  any  body 
is  proportional  to  the  total  quantity  of  matter  which  it  contains. 

Now  suppose  that  two  bodies,  whose  masses  are  equal  in  amount, 
are  moving  in  the  same  straight  line  towards  each  other  with  equal 
speeds.  Each  possesses  an  equal  amount  of  energy,  E  (say).  We 
may  assume  that  the  result  of  the  impact  is  that  each  body  is 
brought  to  rest,  so  that  the  quantity  of  energy  %E  has  been 
expended  in  stopping  the  forward  motion  of  the  two  masses.  Next, 
let  one  of  the  bodies  be  at  rest  while  the  speed  of  the  other  is 
doubled.  The  relative  speed  of  the  two  is  unaltered,  and  hence 
the  energy  expended  in  impact  is  still  ZE.  But  experiment  shows 
that  the  two  bodies  will  now  move  together  with  half  the  speed 
of  the  single  one,  so  that  each  body  still  possesses  energy  E.  There- 
fore the  single  body,  moving  with  double  speed,  possesses  an  amount 
of  energy  4E. 

By  this  experiment,  and  by  other  similar  experiments  conducted 
under  varying  conditions  regarding  the  speeds  of  the  moving  bodies, 


INTRODUCTORY:  THE  PHYSICAL  UNIVERSE.  5 

it  may  be  proved  that  the  energy  of  a  moving  body  varies  directly  as 
the  square  of  its  speed. 

If  E  be  the  energy  of  the  moving  body,  we  may  express  the 
results  just  obtained  by  means  of  the  equation 

E  = 


where  k  is  a  constant.  The  value  of  k  is  purely  arbitrary,  depend- 
ing on  our  choice  of  the  unit  of  energy,  but  it  will  be  shown 
subsequently  that  it  is  convenient  to  choose  the  units,  so  that  the 
value  of  k  is  £.  Thus 

E  =  $mv2. 

8.  Energy,  when  it  is  regarded  as  in  the  preceding  paragraph, 
is  called  energy  of  motion  or  kinetic  energy.  But,  as  already 
indicated,  we  do  not  always  perceive  the  system  to  which  such 
energy  is  communicated.  The  kinetic  energy  which  the  visible 
system  still  possesses  becomes  less  and  less,  and  at  last,  in  a  certain 
position  of  the  system,  it  may  entirely  vanish.  The  kinetic  energy 
remaining  at  any  instant  may  obviously  be  expressed  in  terms  of 
the  position  of  the  system,  and  so  also  may  the  energy  which  has 
been  given  away  from  it.  Therefore,  on  the  assumption,  made  in  §  7, 
that  the  gain  of  energy  of  the  invisible  system  is  equal  to  the  loss 
of  energy  of  the  visible  one,  we.  can  represent  the  kinetic  energy 
of  the  invisible  system  in  terms  of  the  position  of  the  visible  one. 
This  is  always  done  when  the  two  systems  are  so  connected  that, 
when  left  to  themselves,  the  energy  will  be  recommunicated  to  the 
original  one,  and  so  we  speak  of  energy  of  position  or  potential 
energy. 

As  a  special  example  we  may  consider  the  case  of  a  bullet  fired 
vertically  upwards.  The  further  it  rises  the  less  its  speed  becomes, 
and  the  less  work  it  can  do  in  overcoming  obstacles.  At  last  it 
comes  to  rest,  and  is  totally  devoid  of  kinetic  energy.  But  we 
have  only  to  let  it  fall  down  again,  and  (neglecting  the  resistance 
of  the  air)  we  find  that  its  energy  of  motion,  when  it  again  reaches 
the  ground,  has  the  same  value  as  at  first.  Hence,  instead  of 
saying  that  as  the  ball  loses  energy  some  connected  system  gains 
it,  we,  for  convenience,  say  that  as  it  loses  kinetic  energy  it  gains 
potential  energy. 

We  do  not  yet  know  what  this  connected  system,  in  the  case  of 
gravitation,  is.  If  Le  Sage's  hypothesis  of  ultra-mundane  corpuscles 
(Chap.  VIII.)  were  true,  the  kinetic  energy  of  a  ball,  projected 
upwards  against  gravity,  would  be  transmuted  into  kinetic  energy 
of  the  corpuscles. 

We  may  distinguish  a  number  of  forms  of  energy,  all  of  which 


6  A   MANUAL   OF   PHYSICS.  , 

can  be  classified  under  the  two  main  types  just  defined.  Kinetic 
and  potential  energy  of  visible  portions  of  matter  have  been  already 
considered.  There  is  also  kinetic  energy  of  invisible  portions  of 
matter  (Chap.  XX.),  as  in  the  case  of  a  body  which  is  sensibly  hot. 
And  potential  energy  also  exists  on  a  similar  scale,  as  in  the  case  of 
the  so-called  Latent  Heat  (Chap.  XXIII.).  Other  examples  appear  in 
the  molecular  motion  of  gases,  and  in  the  transmission  of  vibrations 
through  an  elastic  medium,  etc.  Again,  two  oppositely  electrified 
bodies  attract  each  other,  and  so  have  potential  energy  of  electrical 
separation.  And,  when  electricity  flows  along  a  conductor,  the  energy 
of  electricity  in  motion  becomes  evident.  Also  two  chemical  sub- 
stances which  tend  to  combine  and  form  a  compound  substance  are 
said  to  have  potential  energy  of  chemical  separation.  Lastly,  two 
magnets  have  potential  energy  relatively  to  each  other,  and  work 
can  be  indirectly  produced  by  means  of  the  motion  of  magnets. 

9.  At  this  point  we  are  led  to  regard  that  characteristic  in  the 
possession  of  which  energy  differs  totally  and  fundamentally  from 
matter.  While  matter  is  essentially  passive,  energy  is  constantly 
in  a  state  of  change.  It  is  constantly  being  handed  on  from  one 
portion  of  matter  to  another,  and  is  ever  being  changed  from  one 
to  another  of  the  forms  above  indicated.  It  is  said  to  possess  the 
property  of  TRANSFORMATION. 

Only  in  virtue  of  this  property  can  we  recognise  its  existence. 
We  could  never  have  known  that  a  moving  cannon  ball  possessed 
energy  had  we  never  seen  its  destructive  effects.  We  would  have 
been  ignorant  of  the  energy  of  an  electrified  thunder-cloud  if  we 
had  not  seen  the  production  from  it  of  light,  heat,  sound,  and 
mechanical  effect.  How  energy  is  passed  on  from  one  material 
system  to  another,  and  how  it  changes  from  one  form  to  another, 
are  questions  to  which  no  final  answer  can  at  present  be  given. 

A  simple  example  of  the  transformation  of  energy  is  furnished  by 
the  motion  of  an  ordinary  pendulum.  At  the  lowest  part  of  the 
swing  the  energy  is  entirely  kinetic  ;  at  the  highest  part  it  is  entirely 
potential ;  and,  in  intermediate  positions,  it  is  partly  kinetic,  partly 
potential. 

A  somewhat  more  complex  example  occurs  in  the  transmission  of 
a  message  by  telephone.  There  is  first  the  energy  of  vibratory 
motion  of  the  air  when  the  sound  is  produced.  This  vibratory 
motion  is  communicated  to  the  metallic  diaphragm  of  the  telephone. 
But  the  diaphragm  is  magnetised  by  induction,  and  so  its  motion 
causes  alterations  in  the  intensity  of  magnetisation  of  the  magnet. 
These  alterations  in  the  magnetisation  produce  electric  currents  in 
the  wire  coiled  round  the  magnet,  and  these  currents  produce  similar 


INTRODUCTORY  :    THE    PHYSICAL   UNIVERSE.  7 

alterations  of  magnetisation  in  the  magnet  of  the  receiving  telephone, 
and  so  similar  motions  of  its  diaphragm  ensue.  Consequently  similar 
sounds  are  heard  at  the  receiving  instrument. 

By  the  consideration  of  such  special  examples  we  are  led  to  the 
conclusion  that  any  form  of  energy  may,  directly  or  indirectly,  be 
changed  into  any  other.  Many  evidences  of  this  will  appear  in 
subsequent  chapters. 

10.  During  all  its  changes  and  transferences  one  thing  is  evident 
regarding  the  energy  in  the  universe — the  total  amount  of  it  is 
unalterable.     This  is  made  clear  by  the  fact  that  strict  '  mechanical 
equivalents'  of  heat  and  the  various  other  forms  of  energy  are 
obtainable  (Chap.  XXV.).  Energy,  like  matter,  possesses  the  property 
of  CONSERVATION.    The  swings  of  a  pendulum,  which  has  been  set  in 
motion  and  then  left  to  itself,  gradually  die  away,  and  finally  vanish. 
But  if  no  energy  were  lost  because  of  the  communication  of  motion 
to  the  air,  and  if  none  were  lost  because  of  friction  at  the  points  of 
support,  or  because  of  vibrations  set  up  in  the  supporting  framework, 
etc.,  the  motion  would  go  on  for  ever.     That  is  to  say,  the  energy 
communicated  to  other  bodies  up  to  any  instant,  together  with  the 
energy  still  possessed  by  the  pendulum  at  that  instant,  is  equal  in 
amount  to  the  original  quantity.     And  the  same  is  true  of  any  other 
system.     Therefore,  since  it  is  conserved,  we  must  regard  energy  as 
having  real  existence. 

11.  A  question  of  the  deepest  importance  to  mankind  arises  in 
connection  with  the  transformation  of  energy.     Are  all  forms  of 
energy  equally  transformable  ?     When  energy  is  changed  from  one 
form  to  another,  can  it  with  equal  readiness  be  changed  back  again 
into  the  original  form  ?     If  not,  it  necessarily  follows  that  the  whole 
amount  of  energy  in  the  universe  will  gradually  assume  that  par- 
ticular  form  which  is  least  transformable.     Observation  and  experi- 
ment have  shown  that  there  is  one  form  into   which  all  others 
are  gradually  and  permanently  changing;  and  that  form  is  the 
energy  of  molecular  motion  known  as  heat.     But  there  is  a  constant 
tendency  towards  diffusion  of  heat,  so  as  to  produce  uniformity  of 
temperature  ;  and  when  uniformity  of  temperature  is  arrived  at,  no 
mechanical  work  can  be  produced  from  the  heat.     The  total  amount 
of  energy  in  the  universe  will  be  the  same  as  before,  in  accordance 
with  the  principle  of  conservation ;  but  none  of  it  will  be  available 
for  the  production  of  mechanical  work.     This  principle  of  the  loss 
of  availability  of  energy  is  technically  known  as  the  principle  of 
DISSIPATION,  or  (preferably  perhaps)  DEGRADATION  of  energy. 

Examples  of  degradation  of  energy  occur  everywhere  in  nature. 
No  stone  falls  from  a  cliff,  no  storm  arises  or  ceases,  no  flash  of 


8  A   MANUAL   OF   PHYSICS. 

lightning  or  peal  of  thunder  occurs,  no  wave  breaks  upon  the  shore, 
without  a  diminution  of  the  possible  amount  of  useful  work  obtain- 
able for  man  by  natural  processes. 

In  accordance  with  this  principle,  potential  energy  of  visible 
portions  of  matter  tends  towards  a  minimum  value,  i.e.,  tends  as  far 
as  possible  to  take  the  form  of  kinetic  energy.  But  further  con- 
sideration of  this  subject  must  be  deferred  in  the  meantime. 

12.  Nothing  which  is  not  either  matter  or  energy  is  conserved — 
at  least,  in  the  same  sense  as  that  in  which  we  assert  conservation 
of  these  things.  Matter  and  energy  are  both  signless  quantities. 
We  might  assert  conservation  of  a  quantity  which  may  be  positive 
or  negative,  provided  that,  when  a  new  positive  amount  of  it  is  pro- 
duced, an  equal  negative  amount  necessarily  appears.  In  this  case 
the  total  algebraic  sum  of  the  quantity  is  constant.  But  if  we 
regard  either  the  positive  portion  of  it  or  the  negative  portion  of  it, 
we  find  that  the  amount  of  either  portion  may  be  perfectly  arbitrary  ; 
and  this  is  not  the  sense  in  which  conservation  is  asserted  of  matter 
and  energy. 

.  In  the  new  sense  alluded  to,  we  speak  of  the  conservation  of 
momentum  (Chap.  VI.),  and  sometimes  even  of  the  conservation  of 
electricity. 


CHAPTEE  II, 

THE   METHODS   OF   PHYSICAL    SCIENCE. 

13.  THE  whole  body  of  scientific  knowledge  has  been  obtained  by 
one  or  other  of  two  methods — observation  or  experiment :  nor  can 
strict  knowledge  be  obtained  in  any  other  way.  The  first  scientific 
investigations  ever  made  must  have  been  of  a  purely  observational 
type,  and  were  very  probably  astronomical  in  their  nature.  In 
making  observations,  we  notice  the  positions  of  objects  and  the 
sequence  of  events ;  and  we  attempt,  then,  to  make  out  relations 
among  them.  In  this  way  arose  the  still-extant  grouping  of  the 
stars  into  various  constellations,  and  —  greatest  perhaps  of  .  all 
examples — the  discovery  by  Kepler  of  the  laws  which  regulate  the 
motions  of  the  planets.  But,  when  we  alter  at  will  the  condi- 
tions attending  certain  phenomena,  so  as  to  discover  the  conse- 
quent alterations  produced  in  the  phenomena,  we  are  said  to  ex- 
periment. It  is  true,  indeed,  that  we  cannot  always  draw  a  hard 
and  fast  distinction  between  observation  and  experiment.  Thus, 
in  calculating  the  speed  of  light  from  observations  upon  the 
satellites  of  Jupiter,  although  we  do  not  ourselves  alter  any  con- 
ditions, yet  we  purposely  take  advantage  of  alterations  which  occur 
naturally. 

By  such  means  we  first  of  all  obtain  mere  series  of  fads  often 
without  any  mutual  connection  whatsoever,  and,  not  infrequently, 
so  grouped  as  to  suggest  false  relations.  The  next  duty  of  the 
scientist  is  to  group  these  isolated  data  after  a  definite  system,  to 
co-ordinate  the  facts  with  the  object  of  subsequently  discovering  the 
true  relations  which  subsist  among  them ;  and  the  greater  the 
power  of  the  observer  to  detect  real  resemblances  and  essential 
differences  the  sooner  will  his  ulterior  object  be  attained. 

The  question  of  cause  and  effect  next  arises.  Of  two  phenomena 
which  appear  successively,  and  no  one  of  which  appears  without  the 
other,  that  one  which  is  first  evident  is  usually  called  the  cause  of 
the  other,  which  is  said  to  be  its  effect.  But,  obviously,  great  care 


10 


A  MANUAL   OF   PHYSICS. 


must  be  taken  to  avoid  any  error  in  such  an  assertion,  for  there  may 
be  many  sources  of  mistake.  In  the  first  place,  it  is  conceivable  that 
two  phenomena  might  appear  in  invariable  succession  the  one  to 
the  other,  and  yet  the  true  explanation  might  be  that  they  had  a 
common  cause,  and  were  not  otherwise  connected.  The  flash  of 
forked  lightning  and  the  sound  of  thunder  occur  successively ;  but 
the  sound  is  due  to  the  explosive  expansion  of  the  air  heated  by  the 
passage  of  the  electricity,  while  a  portion  of  the  light  is  also  caused 
by  this  explosive  expansion,  which  compresses  the  adjacent  layers 
so  suddenly  as  to  render  them  luminous  by  the  excessive  heat  so 
developed.  Again,  the  occurrence  of  one  event  is  frequently  neces- 
sary, in  order  that  we  may  perceive  another  between  which  and  the 
former  there  is  no  connection  whatsoever;  and,  frequently,  the 
effect  becomes  evident  before  the  cause  is  noticed.  Still  further,  we 
observe  that  events  sometimes  occur  simultaneously.  Thus,  tornadoes 
are  often  due  to  the  sudden  heating  of  large  portions  of  the  atmo- 
sphere by  means  of  the  latent  heat  given  out  on  rapid  condensation 
of  vapour.  But  the  condensation  of  vapour  and  the  evolution  of 
latent  heat  occur  of  necessity  at  one  and  the  same  instant,  so  that 
we  might  with  equal  propriety  refer  the  tornado  to  either  event  as  a 
cause. 

In  nature  there  is  an  apparently  endless  series  of  causes.  Each 
event  is  the  cause  of  another,  and  was  itself  produced  as  the  conse- 
quence of  a  preceding  event.  When  a  large  mass  of  cloud  intercepts 
the  rays  of  the  sun  from  the  underlying  atmosphere,  the  air  grows 
colder,  and  as  it  grows  colder  it  contracts.  This  causes  an  inrush 
of  air  from  surrounding  regions,  which  well-known  result  is  expressed 
by  the  popular  phrase  that  the  cloud  or  the  rain  '  draws '  the  wind. 
The  effects  of  this  motion  might  be  traced  out  endlessly  if  our  senses 
were  sufficiently  acute  and  our  powers  sufficiently  universal ;  and 
so  also  the  various  motions  preceding  the  motion  of  the  cloud  might 
be  traced. 

14.  Among  his  other  duties,  the  physicist  has  to  undertake  the 
investigation  of  the  effects  which  result  from  physical  conditions. 
Such  an  investigation  is  comparatively  simple.  He  has  only  to 
make  certain  that  the  effects  which  he  observes  are  not  due  to  any 
unnoticed  conditions.  The  converse  problem — the  investigation  of 
causes — is  not  by  any  means  so  simple.  The  investigator  must  first 
determine  the  various  physical  conditions  which  actually  obtain, 
and  he  must  then  find  out  which  of  these,  if  any,  are  essential  to 
the  production  of  the  phenomenon.  If  three  conditions  are  observed, 
an  experiment  must  be  made  in  which  all  of  them  are  present,  in 
order  to  make  it  certain  that  the  result  really  follows.  Then  three 


THE    METHODS    OF   PHYSICAL   SCIENCE.  11 

experiments  must  be  made  with  the  three  pairs  of  conditions. 
Three  more  must  then  be  performed  with  each  condition  present 
alone.  Lastly,  it  may  be  necessary  to  make  another  experiment 
in  the  absence  of  all  the  conditions.  In  all,  eight  experiments 
may  be  necessary  when  there  are  three  conditions.  If  four  con- 
ditions are  present,  one  experiment  with  all  the  conditions  present, 
four  with  three  conditions,  six  with  two,  four  with  one,  and  one 
with  none,  may  be  required — in  all  sixteen  experiments.  With 
one  condition  only,  not  more  than  two  experiments  are  necessary, 
and  the  number  is  doubled  for  every  additional  condition  introduced. 
If  only  ten  conditions  existed,  more  than  a  thousand  experiments 
would  be  necessary  to  completely  exhaust  all  the  possibilities. 
Obviously,  science  could  make  little  progress  were  such  immense 
labour  a  necessity.  Fortunately  it  is  not.  Past  experience  and 
natural  instinct  indicate  to  the  experimenter  the  direction  in  which 
truth  lies,  and  thus  he  is  often  enabled  to  take  a  short  road  to  the 
end  in  view. 

15.  One  great  means  by  which  labour  is  reduced  is  the  employ- 
ment of  a  suitable  and  probable  hypothesis.  Certain  facts  are 
known,  and  a  hypothesis  is  framed  regarding  their  explanation. 

The  greater  the  number  of  phenomena  which  a  given  hypothesis 
can  explain,  the  greater  is  the  likelihood  of  its  truth.  When  only 
some  facts  fall  in  with  the  assumptions  while  others  do  not,  some 
modification  of  the  hypothesis  must  be  made.  But  when  new 
modifications  have  to  be  made  for  every  new  requirement,  it  is  time 
to  abandon  the  hypothesis  and  seek  for  another  and  more  probable 
one.  A  good  hypothesis  must  explain  all  the  facts  for  the  elucida- 
tion of  which  it  was  framed.  It  should  also  explain  other  known 
facts,  and  facts  which  subsequently  become  known.  But,  in  such  a 
case,  it  is  customary  to  speak  of  it  as  a  theory.  Above  all,  a  good 
theory  should  lead  to  the  prediction  of  previously  unknown  facts. 

It  sometimes  happens  that  different  theories  are  each  sufficient  for 
the  explanation  of  known  facts.  One  theory  may  explain  certain 
phenomena  more  easily  than  another  can,  while  in  the  explanation 
of  others  it  is  more  laboured.  The  logical  consequences  of  the  two 
theories  must  then  be  worked  out  as  far  as  possible,  and  it  will 
usually  be  found  that  at  one  or  more  points  each  leads  to  an  opposite 
conclusion.  Here  experiment  must  step  in  to  determine  which 
conclusion  is  correct,  and  so  to  decide  between  the  two  theories. 
This  experimental  investigation  is  termed  a  crucial  test.  Very 
prominent  examples  occur  in  the  theories  of  heat  and  light. 

The  tendency  of  scientific  investigation  in  the  present  day  is 
towards  the  formation  of  dynamical  explanations  of  all  phenomena — 


12  A   MANUAL   OF   PHYSICS. 

towards  the  production  of  theories  in  which  all  purely  physical 
phenomena  are  explained  in  terms  of  matter,  and  the  energy  which 
is  associated  with  it. 

Mathematical  theories  form  an  important  class  in  which  the 
mathematical  consequences  of  the  fundamental  assumptions  are 
rigidly  worked  out.  When  the  postulates  are  merely  expressions  of 
known  facts,  the  consequences  of  such  theories  may  be  regarded  as 
strictly  true  ;  but  in  making  such  a  statement,  we  must  remember 
that  all  our  knowledge  is  only  approximate,  being  limited  by  the 
imperfections  of  our  senses  and  our  instruments,  so  that  the  above 
expression, '  strictly  true,'  means  merely  that  we  cannot  detect  devia- 
tions from  the  truth.  The  theory  of  gravitation  is  of  this  kind,  and 
it  furnishes  us  with  one  of  the  finest  examples  of  prediction.  From 
irregularities  in  the  motion  of  the  planet  Uranus,  Adams  and 
Leverrier  were  led  to  foretell  the  existence  and  indicate  the  position 
of  the  previously  unknown  planet  Neptune. 

In  other  mathematical  theories  there  is  merely  a  partial  experi- 
mental basis;  for  example,  the  dynamical  theory  of  heat  or  the 
undulatory  theory  of  light.  Again,  it  is  possible  to  work  out  mathe- 
matical theories  of  phenomena  in  which  we  know  that  something 
moves ;  but  we  may  not  know  what  is  moving  or  how  the  motion 
is  propagated.  The  theories  of  heat-conduction  and  of  electro-dyna- 
mics are  prominent  examples. 

As  knowledge  advances  theory  must  cease.  Some  theories  will 
be  shown  to  be  false,  while  the  truth  of  others  will  be  confirmed, 
in  which  case  they  of  necessity  vanish  as  theories. 

A  very  important  scientific  method,  which  is  in  essence  hypo- 
thetical, is  known  as  the  argument  from  analogy.  When  we 
perceive  resemblances  between  different  physical  systems  or  pro- 
cesses, we  say  that  they  are  analogous ;  and  when  any  new  fact  is 
discovered  regarding  one  of  the  systems,  we  are  led  by  analogy  to 
look  for  something  similar  in  the  others.  The  principle  is  of  extreme 
importance  in  experimental  work,  as  it  indicates  a  promising  direc- 
tion for  research,  and  so  prevents  aimless  and  often  fruitless  labour ; 
and,  further,  the  failure  of  an  analogy  may  be  as  instructive  as  its 
success.  There  are  many  analogies  between  the  phenomena  of 
sound  and  of  light ;  but  there  is  nothing  in  sound  which  corresponds 
to  polarisation  in  light,  and  the  distinction  is  of  fundamental  im- 
portance. 

Another  extremely  important  aid  to  research  is  derived  from  the 
condition  for  stable  equilibrium.  This  condition  may  be  expressed 
as  follows :  A  system  is  in  stable  equilibrium,  under  given  physical 
conditions,  when  any  small  variation  of  one  or  more  of  these  pro- 


THE    METHODS    OF    PHYSICAL    SCIENCE.  13 

duces  other  variations  which  would  themselves,  as  causes,  produce 
changes  opposite  to  the  first. 

[It  is  easy  to  see  that  this  statement  does  express  the  condition 
for  stable  equilibrium.  For,  if  one  variation  produced  another 
variation  which  caused  further  variation  of  the  first  kind,  this 
additional  variation  would  cause  more  variation  of  the  second  kind, 
and  so  on  reciprocally.  Therefore  the  variation,  once  started,  would 
constantly  increase,  or,  in  other  words,  the  presumed  state  of 
equilibrium  is  unstable.] 

The  equilibrium  of  a  body  supported  by  the  hand  affords  a  ready 
illustration.  Increased  pressure  of  the  hand  upon  the  body  causes 
an  upward  motion  of  the  body.  Conversely,  the  independent  com- 
munication of  upward  motion  to  the  body  diminishes  the  normal 
pressure  between  the  hand  and  it. 

Another  example  is  furnished  by  water,  which  is  physically 
stable  (under  ordinary  circumstances)  below  its  maximum-density 
point,  and  which,  at  temperatures  below  its  maximum- density  point, 
contracts  when  the  temperature  is  raised.  Hence,  in  accordance 
with  the  above  principle,  we  can  assert  that  sudden  diminution  of 
volume  caused  by  the  application  of  pressure  will  produce  a  fall  in 
temperature.  Again,  sudden  elongation  heats  indiarubber;  there- 
fore the  heating  of  the  stretched  indiarubber  makes  it  shrink.  Sir 
W.  Thomson  has  proved  experimentally  that  both  these  results  are 
true.  We  shall  see  subsequently  that  they  follow  as  consequences 
of  the  dynamical  theory  of  heat. 

16.  In  all  observations,  alike  of  natural  processes  and  of  experi- 
mental results,  errors  of  observation  are  almost  certain  to  arise. 
Such  inaccuracies  are  as  likely  to  be  in  excess  as  in  defect,  and  are 
much  more  likely  to  be  small  than  to  be  large,  while  a  very  large 
error  will  practically  never  occur  at  all  unless  the  method  of  obser- 
vation is  an  extremely  objectionable  one.  To  get  rid  of  these 
errors  we  must  make  a  sufficient  number  of  independent  observa- 
tions. In  any  one  observation  we  do  not  expect  the  result  to  be 
correct ;  but  there  is  a  certain  numerical  quantity,  called  the  pro- 
bable error,  such  that  the  actual  error  is  as  likely  to  be  greater  than 
it  as  to  be  smaller  than  it.  If  each  observation  is  made  under  con- 
ditions precisely  similar  to  those  of  another,  the  probable  error 
of  each  is  the  same.  In  this  case  we  simply  take  the  arithmetical 
mean  of  all  the  observations,  and  this  gives  the  result  which  is 
most  likely  to  be  near  the  true  value.  But  if  each  observation  is 
not  made  under  precisely  similar  circumstances,  the  probable  error 
of  each  will  in  general  be  different,  and  its  value  will  be  known  for 
each  from  the  known  experimental  conditions.  The  most  probable 


14  A   MANUAL    OF   PHYSICS. 

value  is  now  found  by  the  method  of  least  squares.  As  a  simple 
example,  let  us  take  the  case  of  three  independent  observations  of  a 
quantity  #,  which  gives  the  results  x  =  a,  x  =  b,  x  —  c:  and  let  the 

probable  error  of  b  and  c  be  -  th  and  —  th  of  that  of  a  respectively.    If 

we  now  multiply  the  second  and  third  equations  by  n  and  m  respect- 
ively, we  get  x  =  a,  nx  =  nb,  mx  =  mc,  where  the  probable  error  of 
the  right-hand  member  of  each  equation  is  the  same.  Multiply  again 
by  n  and  m  as  before,  and  we  get  x  =  a,  n-x  =  n2b,  m~x  =  ni-c.  These 
equations  give 

_  a  +  n2b  •+-  m-c  ' 
x  —  _ — -  — - — - —  -—  j 
1  +  n-  -f-  m2 

a  value  which  makes  the  sum  of  the  squares  of  the  errors  of  the 
original  equations  a  minimum.  If  the  equations  contain  more  than 
one  quantity  subject  to  error,  the  same  method  applies,  for  the 
number  of  observations  will  usually  be  much  greater  than  the 
number  of  unknown  quantities. 

In  addition  to  errors  of  observation  there  may  be  errors  which 
tend  always  in  one  direction,  so  that  the  result  obtained  is  either  too 
large  or  too  small.  Such  errors  are  due  usually  to  the  instrument 
or  to  the  method  of  observation  used,  and  are  generally  termed 
instrumental  errors.  Under  this  heading  may  be  included  errors 
due  to  peculiarities  of  the  observer,  and  the  correction  to  be  applied 
is  termed  the  personal  equation.  When  only  comparative  values 
of  a  quantity  under  different  circumstances  are  required,  such  errors 
frequently  affect  each  observation  alike,  and  so  may  be  neglected. 
But,  in  general,  they  must  be  eliminated  by  varying  the  instrument 
and  the  observer  and  combining  the  results  as  above. 

17.  Most  frequently  in  physical  inquiries  we  have  to  investigate 
the  variations  of  some  quantity  consequent  upon  the  variation  of 
another.  The  experiment  may  often  be  so  arranged  as  to  give  a  con- 
tinuous record  of  the  mutual  variation  of  the  two,  as  in  the  case 
of  the  self -registering  thermometer  and  similar  instruments  ;  or 
even  a  simultaneous  and  continuous  record,  as  in  the  case  of  the 
rise  of  water  in  the  wedge-shaped  space  between  two  vertical  glass 
plates  (§  122).  But,  more  generally,  the  results  of  a  few  separate 
experiments  are  given,  each  of  which  records  one  definite  value  of 
the  one  quantity  corresponding  to  one  definite  value  of  the  other. 
From  these  detached  results  the  law  connecting  the  variations  of  the 
two,  or,  rather,  an  approximate  law  must  be  found,  the  approxima- 
tion to  be  so  exact  that  the  result  given  by  the  law  for  intermediate 
values  of  the  quantities  shall  not  differ  from  that  which  may  be 
determined  afterwards  by  experiment  by  an  amount  greater  than 


THE   METHODS   OF   PHYSICAL  SCIENCE.  15 

the  possible  error  of  observation.  Such  a  relation  is  termed  an 
empirical  law,  and  the  formula  expressing  it  is  called  an  empirical 
formula. 

The  formula 

y  =  a  +  b  (a?  -BO)  +  c  (x  -  x0)  2  -f  ----  , 

is  frequently  adequate  for  the  close  representation  of  many  experi- 
mental results.  The  constant  a  is  the  observed  value  of  y  when 
x  =  x0;  and  the  values  of  the  constants  &,  c,  etc.,  are  found  from  a 
series  of  particular  equations  obtained  from  the  above  by  giving  x 
and  y  simultaneously-observed  numerical  values.  Frequently  no 
more  than  three  terms  are  necessary.  For  example,  the  values  of 
x  and  y  which  are  contained  in  the  table  below  are  accurately 
represented  by  the  formula 

T/  =  1+2  (a  -1)  +  (z-l)2, 
or  by 

y  =  9  +  6  (»-3)  +  (x  -3)2,  etc. 

If  we  have  obtained  by  experiment  n  values  of  y  corresponding  to 
n  values  of  x,  we  may  use  as  an  empirical  formula  the  equation 
(given  by  Laplace)  : 

-£-  -1 

x  -  XL  (xl  -  x2)(Xl  -  a?8)  .  .  .  .  (ajj  -xn) 


This  obviously  gives  y  =  yv  when  x  =  xlt  etc.,  so  that  all  the  observed 
values  are  accounted  for. 

When  values  of  y  corresponding  to  equi-different  values  of  x  are 
observed,  the  symbolical  equation 


is  specially  useful.  Here  m  means  '  the  numerical  value  of  y, 
which  stands  mth  in  the  observed  series';  Am  =  m+T--m;  and 
A2m=  Am-j-1-  Am.  Am,  etc.,  are  called  the  '  first  differences,' 
and  A2m,  etc.,  ?re  called  the  'second  differences'  of  the  observed 
values  of  y.  For  example,  let  the  values  of  x  be  the  natural  num- 
bers, while  the  values  of  y  are  the  squares  of  these.  From  the 
tabulated  results : 

x  1234567 

y  1        4        9        16        25        36        49 
Ay  3       5        7          9         11        13 

A2?/  22222 


16 


A   MANUAL    OF    PHYSICS. 


we  get,  by  the  formula,  for  the  sixth  observed  value  of  y  the  quan- 
tity 6  =  3+3  =  3+3A3+3A23  =  9+3x7+3x2  =  36;  Or  6  =  4+2  = 
4+2A4~+A24  =  16+2x9+2  =  36;  and  so  on. 

When  the  observed  values  are  sufficiently  close  together,  such 
formulae  enable  us  to  find  intermediate  values  with  considerable 
accuracy,  and  also  to  find  values  altogether  outside  the  experimental 
range  for  a  short  distance ;  but,  if  pushed  too  far,  the  formulae  will 
give  values  differing  more  and  more  from  the  truth. 
jj;  Instead  of  seeking  for  an  empirical  formula,  we  might  plot  a 


FIG. 


curve,  the  abscissae  of  which  represent  the  values  of  the  quantity 
to  which  we  give  arbitrary  values,  while  the  ordinates  represent  the 
values  of  the  quantity  whose  variation  we  observe.  The  points  so 
obtained  will  not  generally  lie  on  a  smooth  curve  because  of  observa- 
tional errors,  but  a  curve  drawn  freely  through  them,  leaving  on 
the  whole  as  many  points  on  one  side  of  it  as  there  are  on  the  other, 
will  be  fairly  free  from  such  inaccuracies,  and  will  generally  give  a 
better  approximation  to  the  truth  than  the  actual  points  themselves 
give.  This  is  known  as  the  graphical  method,  and  is  largely  used 


THE    METHODS    OF    PHYSICAL    SCIENCE.  17 

I 

by  experimentalists.  For  example,  the  square  of  the  time  of  oscil- 
lation of  a  simple  pendulum  is  proportional  to  the  length  of  the  pen- 
dulum. Hence,  if  we  make  the  abscissae  represent  lengths  while 
the  ordinates  represent  the  squares  of  the  periods  of  oscillation,  we 
should  get  a  straight  line  passing  through  the  origin.  In  Fig.  1, 
which  is  drawn  from  the  results  of  actual  experiment,  it  will  be 
seen  that  the  points,  while  they  all  agree  very  well  with  each  other, 
do  not  give  the  proper  ratio  of  the  quantities.  The  straight  line  is 
inclined  at  the  proper  angle.  From  the  close  agreement  of  the 
different  results,  we  infer  that  there  is  little  error  of  observation, 
but  the  wrong  inclination  of  the  line  shows  that  there  must 
be  an  instrumental  source  of  error.  If  a  series  of  experiments 
were  made  at  parts  of  the  earth's  surface  where  gravity  had 
sensibly  different  values,  we  should  get  a  series  of  straight  lines  all 
passing  through  the  origin,  but  all  inclined  at  different  angles.  All 
such  series  of  curves  may  be  regarded  as  contours  of  a  surface,  and 
this  subject  is  of  such  importance  in  physics  as  to  merit  discussion 
in  a  separate  chapter. 

Very  often  the  form  of  the  curve  obtained  by  the  graphical  method 
indicates  at  once  a  suitable  empirical  formula. 


CHAPTEE  III. 

THE    THEORY   OF   CONTOURS,    AND   ITS   PHYSICAL   APPLICATIONS. 

18.  THE  nature  of  any  quantity  is  completely  known  when  it  is 
understood  ivhat  units  are  involved  in  its  measurement,  and  how 
they  are  involved.  Thus  a  speed  involves  the  unit  of  length 
directly  >  and  the  unit  of  time  inversely ;  an  acceleration  involves 
a  length  directly,  and  the  square  of  a  time  inversely.  But,  when 
dealing  with  extension,  we  have  only  to  consider  the  unit  of  length. 
We  say  that  the  extension  under  consideration  has  one,  two,  or 
three,  etc.,  dimensions,  according  as  the  unit  of  length  is  involved  to 
the  first,  second,  or  third,  etc.,  power.  A  line  has  only  one  dimen- 
sion, for  only  one  number,  with  the  proper  sign  attached,  is  required 
to  completely  specify  the  relative  position  of  two  points  on  the  line. 
A  surface  has  two  dimensions,  for  two  directed  lengths  define  the 
position  of  a  point  on  it  with  reference  to  any  other  point  taken  as 
origin.  Thus  we  speak  of  one  point  on  the  surface  of  the  earth  as 
being  so  much  north  or  south,  and  so  much  east  or  west  of  another. 
Three  directed  lengths  determine  the  relative  position  of  two  points 
in  space,  that  is,  in  extension  of  the  third  order.  Thus  we  speak  of 
the  length,  breadth,  and  thickness  of  a  solid. 

It  must  be  observed  that  three  lengths  are  not  necessary.  One 
of  the  given  conditions  must  be  a  length,  but  the  others  might  be 
angles.  For  example,  instead  of  saying  that  one  mountain-top  is 
so  far  north  or  south,  so  far  east  or  west,  and  so  much  higher  or 
lower,  than  another,  we  might  give  the  distance  between  the  two 
peaks  and  their  relative  altitude  and  azimuth. 

The  intersection  of  any  surface,  which  has  a  constant  charac- 
teristic, with  the  surface  of  a  solid  is  called  a  contour-line.  A  good 
illustration  is  furnished  by  the  contour-lines  in  an  Ordnance  map. 
Any  such  line  is  the  intersection  of  a  surface,  all  points  of  which 
are  at  a  constant  height  above  sea -level,  with  the  surface  of  the 
solid  earth.  An  Ordnance  map  gives  a  very  good  idea  of  the  dis- 
tribution of  places  on  the  earth's  surface  as  regards  height,  as  well 


THE    THEORY    OF    CONTOURS,    AND    ITS    PHYSICAL   APPLICATIONS.       19 

as  with  reference  to  latitude  and  longitude,  and  the  information  is 
more  and  more  minute  as  the  number  of  lines  is  greater.  In  other 
words,  contours  enable  us  to  represent  on  a  plane  surface  the  mutual 
relations  of  three  quantities. 

This  gives  the  key  to  the  great  importance  of  the  theory  of  con- 
tours in  physical  science.  For,  if  the  physical  condition  of  a  substance 
is  completely  denned  when  the  simultaneous  values  of  three  of  its 
properties  are  given,  we  can  construct  a  solid  the  surface  of  which 
represents  all  possible  conditions  of  the  substance — just  as  we  can 
construct  a  model  of  the  earth's  surface  in  terms  of  latitude, 
longitude,  and  height  above  sea-level. 

19.  We  may  extend  the  above  conception,  and  state,  generally, 
that  the  contour  of  an  object  o/n  dimensions,  existing  in  extension 
of  the  (n+l)th  order,  is  its  intersection  with  an  object  of  n  dimen- 
sions at  every  point  of  which  some  quantity  has  a  constant  value. 
It  is,  therefore,  of  (n  —  1)  dimensions. 

Since,  in  extension  of  the  (n+l)th  order,  we  may  have  objects  of 
less  dimensions  than  the  nth,  we  might  make  the  further  develop- 
ment that  in  extension  of  the  (n+l)th  order  we  may  have  contours 
of  all  positive  dimensions  up  to  the  (n  -  l)th  inclusive.  Indeed, 
there  is  no  reason  why  we  should  not  consider -contours  of  n  dimen- 
sions in  space  of  (n  +  1)  dimensions.  Hence,  in  ordinary  extension, 
we  may  have  point,  curve,  and  surface  contours.  The  contours  of 
a  curve  are  points ;  of  a  surface,  curves ;  of  a  solid,  surfaces ;  of 
a  four-dimensional  object,  solids  ;  and  so  on. 

The  properties  of  four- dimensional  extension,  or  even  of  exten- 
sion of  the  nih  order,  can  be  treated  mathematically ;  but,  from 
want  of  experience,  it  is  impossible  to  imagine  the  nature  of  such 
extension. 

20.  By  means  of  contour-points  the  nature  of  curves  may  be 
exhibited  in  diagrams  consisting  of  straight  lines  only.      For  we 
may  intersect  a  given  curve  by  curves,  along  each  of  which  some 
quantity  has  a  constant  value,  and  then  project  the  points  of  inter- 
section upon  any  straight  line. 

Consider  first  a  plane  curve,  and,  for  convenience,  let  its  plane  be 
taken  as  that  in  which  two  co-ordinate  quantities,  x  and  y,  are 
measured  in  perpendicular  directions  from  the  same  origin.  Let 
fn  (xi  y)  —  °  foe  the  equation  of  the  given  curve,  where  the  suffix 
denotes  the  degree  of  the  equation.  The  equations  to  the  curves 
along  which  some  quantity,  say  c,  is  constant,  may  be  written  in  the 
general  form,  0n  (x,  y,  c)  =  o.  In  the  different  curves  of  the  system 
c  has  different  values.  As  a  particular  example,  the  curves  might 
be  circles  of  different  radii.  Again,  the  equation  might  be 

2—2 


20  A   MANUAL   OF   PHYSICS., 

<£i  (y>  c)  =  o,  which  represents  a  series  of  lines  parallel  to  the  axis  of 
sc.  This  is  the  simplest  case  which  we  can  consider,  and,  at  the 
game  time,  the  most  useful.  The  curves  in  Figs.  2  and  3  are  inter- 
sected by  lines  parallel  to  the  axis  of  x,  and  the  points  of  inter- 
section are  projected  upon  the  axis,  and  are  designated  by  numbers 
which  give  the  various  values  of  y  corresponding  to  the  given  values 
of  x.  If  the  curve  be  continuous,  a  maximum  or  minimum  value 
of  y  exists  between  two  equal  values.  It  is  a  maximum  if  y  first 
increases  and  then  diminishes  as  x  increases  continuously  from  its 
least  value  corresponding  to  the  given  value  of  y.  It  is  a  minimum 
if  y  first  diminishes  and  then  increases.  The  steepness  of  slope  is 
shown  by  the  closeness  of  the  contours  for  equal  increments  of  y, 
and  its  .direction  is  shown  by  the  order  in  which  the  values  of  y 
occur  as  regards  numerical  magnitude  when  x  increases. 


FIG.  2.  FIG.  3. 

In  the  case  of  tortuous  curves  we  may  obtain  the  contours  most 
conveniently  by  cutting  the  curve  by  surfaces  over  which  some 
quantity  is  constant.  In  particular,  these  surfaces  may  be  planes 
perpendicular  to  the  z  axis,  in  which  case  the  equations  are  of  the 
form/j  (z,  c)  =  o. 

The  position  of  a  moving  point  in  space  is  obviously  represent- 
able  by  a  tortuous  curve.  Its  position  at  any  time  can  be  got  from 
the  curve  if  the  value  of  the  time  in  terms  of  one  of  the  co-ordi- 
nates, say  z,  is  known ;  for  we  should  then  only  have  to  cut  the 
curve  by  the  plane  \fn  (z,  t)  =  o,  where  t  represents  the  time,  and 
ifn  is  a  functional  symbol,  showing  that  the  equation  is  of  the  first 
degree  in  z,  but  may  be  of  any  degree  in  t.  [This  condition  is 
rendered  necessary  by  the  fact  that  the  point  must  be  in  one  definite 


THE   THEORY   OF   CONTOURS,   AND   ITS   PHYSICAL  APPLICATIONS.      21 

position  at  a  given  time,  but  may  occupy  the  same  position  at 
different  times.]  If  a  number  of  such  curves  are  simultaneously 
traced  out  in  space  by  moving  material  points,  we  can  obtain  the 
diagram  of  configuration  of  the  material  system  at  any  time  by 
cutting  the  curves  by  planes  corresponding  to  that  time,  and  pro- 
jecting the  points  of  intersection  upon  the  parallel  co-ordinate  plane. 

It  is  evident  that,  in  general,  the  plane  which  corresponds  to  a 
definite  time  will  be  different  for  each  curve.  The  disadvantage  sd 
entailed  may  be  got  rid  of  by  the  employment  of  trilinear  co-ordi- 
nates to  indicate  the  position  of  the  point  when  the  time  is  given; 
If  the  curve  be  cut  by  any  plane,  the  distances  of  the  point  of 
intersection  from  three  intersecting  straight  lines  in  that  plane  give 
the  a?,  y,  z  co-ordinates  at  the  corresponding  instant.  The  value  of 
the  time  might  be  given  by  the  distance  of  the  plane  from  a  fixed 
plane  parallel  to  it. 

The  curves  which  are,  on  this  system,  taken  to  represent  the    ; 


FIG.  4. 

positions  of  the  points  are  not  in  general  the  actual  curves  traced  out 
by  the  points  in  their  motion  through  space.  But  the  diagram  of 
configuration  obtained  from  them  has  the  advantage  of  showing  af 
once  the  values  of  all  the  co-ordinates  of  all  the  points,  whereas, 
in  the  Cartesian  system,  this  could  not  be  done  without  projection 
on  all  the  co-ordinate  planes.  The  triangles  of  reference,  though 
they  may  be  similar,  are  not  generally  of  the  same  magnitude.  A; 
difference  in  magnitude  is  necessary,  in  order  to  represent  varying 
values  of  the  co-ordinates.  In  Fig.  4  the  triangles  are  similar; 
and  equidistant,  while  the  point  a  is  fixed ;  hence  the  diagram  re- 
presents the  linear  motion  of  a  point.  In  general  there  must  be  a 
different  set  of  triangles  for  each  separate  point  whose  motion  is 
to  be  indicated. 

By  the  aid  of  such  a  diagram,  the  diagram  of  total  displace- 
ments (§  40)  in  a  given  time  may  be  constructed.  And  by  taking 
the  displacements  in  one  nth  part  of  the  unit  of  time  (n  being 


22 


A    MANUAL    OF    PHYSICS. 


indefinitely  large),  and  magnifying  them  n  times,  the  diagram  of 
velocities  can  be  got.  Similarly,  the  diagrams  of  accelerations, 
forces,  and  so  on,  may  be  represented  as  the  contours  of  curves. 
The  curves,  from  which  the  diagrams  of  velocities,  etc.,  are  obtained, 
are,  it  is  almost  needless  to  remark,  different  from  the  original 
curves  which  represent  the  positions  of  the  moving  points.  In  the 
case  of  velocities,  they  are  the  hodographs  (§  48)  of  the  original 
curves  on  this  trilinear  system  of  reference. 

As  another  example  of  the  use  of  tortuous  curves,  we  may  con- 
sider two  quantities,  x  and  y,  connected  by  the  equation  if  =  ax, 
which  gives  ydy/dx  =  aj^  (§  30).  We  may  now  take  dyjdx  as  a 
third  co-ordinate  quantity,  and  so  obtain  a  tortuous  curve.  For 
example,  if,  in  this  case,  y  represents  the  time  during  which  a  body 


FIG. 


b. 


has  been  falling  from  rest  under  gravity,  and  if  x  represents  the 
space  described  from  rest,  the  velocity  acquired  is  represented  by 
the  reciprocal  of  the  third  co-ordinate  quantity. 

21.  If  any  curve  be  cut  by  planes  parallel  to  that  of  (x,  y),  and 
if  the  various  points  of  intersection  be  projected  on  any  one  of  these 
planes,  say  z  =  o,  the  contour-points  so  obtained  will  evidently  lie  on 
a  definite  line,  and  the  line  will  be  more  accurately  indicated  in 
proportion  as  the  number  of  intersecting  planes  is  greater  and  their 
mutual  distance  is  less.  It  will  be  given  without  any  break  of  con- 
tinuity by  projecting  every  point  of  the  curve  upon  the  plane  z  —  o. 
But  such  a  line  may  be  regarded  as  the  intersection,  by  the  plane 
z  =  o,  of  a  cylindrical  surface  whose  generating  lines  are  parallel  to 
the  £-axis  and  are  drawn  from  the  given  curve  to  meet  that  plane. 
Now  this  satisfies  our  definition  of  a  contour-line,  for  it  is  the 


THE    THEORY    OF    CONTOURS,    AND    ITS    PHYSICAL   APPLICATIONS.       23 

intersection  of  a  given  surface  by  a  surface  over  which  z  is  constant 
(zero).  A  cylindrical  surface  supplies  the  simplest  diagram  of  con- 
tour-lines. The  contours  are  all  superposed  in  the  diagram,  but 
are  not  in  general  conterminous.  The  only  case  in  which  they 
would  be  conterminous  is  (Fig.  5)  that  in  which  the  same  values  of 
the  x  and  y  co-ordinates  of  a  point  on  a  curve  correspond  to  different 
values  of  z. 

In  the  case  of  a  non- cylindrical  surface,  no  part  of  the  contours 
will  be  superposable  in  general.  The  contours  of  a  hemisphere,  for 
example,  are  concentric  circles  (Fig.  6).  And,  just  as,  in  the  case  of 
contour-points,  the  steepness  of  slope  of  the  curve  is  indicated  by 
the  closeness  of  the  contour-points  on  the  #-axis  for  equal  incre- 
ments of  ?/,  so,  in  the  case  of  contour-lines,  the  steepness  of  slope 
of  the  surface  is  indicated  by  th6  closeness  of  the  contour -lines  for 


FIG.  7.  FIG.  8. 

equal  increments  of  z.  The  contours  are  closer  when  their  radii 
are  large. 

The  contours  of  a  right  circular  cone  are  also  concentric  circles, 
but  they  are  at  equal  distances  apart  for  equal  increments  of  «. 

22.  As  an  example  of  a  surface,  the  contours  of  which  may  be 
used  to  indicate  certain  physical  properties,  we  may  consider  that 
one  whose  equation  is 

ft****!/* 

If  y  represents  the  length  of  a  simple  pendulum,  while  x  represents 
the  square  of  its  time  of  oscillation,  we  know  that  z  represents  the 
value  of  the  acceleration  due  to  gravity.  The  surface  (Fig.  7)  may 
obviously  be  supposed  to  be  produced  by  the  motion  of  a  straight 
line  which  intersects  the  2-axis  and  is  always  perpendicular  to  it, 


. 

24  A   MANUAL    OF    PHYSICS. 


and  which  rotates  uniformly  about  that  axis  while  it  moves  at  a 
constant  rate  along  it.  The  contours  (by  planes  perpendicular  to  the 
axis  of  &)  are  straight  lines  passing  through  the  origin  and  variously 
inclined  to  the  a?-axis  (§  17). 

Again,  the  intrinsic  equation  of  the  circle  is 

S  =  &0, 

where  a  is  the  radius  and  $  is  the  angle  between  the  radius  vector 
-and  the  initial  line.  Hence  the  intrinsic  equation  of  one  involute 
is 


This  involute  is  the  one  which  meets  the  circle  at  the  position 
from  which  <t>  is  reckoned,  and  s'  is  measured  along  it  from  this 
point.  If  we  consider  a  to  be  the  mass  of  a  moving  body,  and  <j>  to 
be  its  speed,  s  and  s'  are  respectively  its  momentum  and  kinetic 
energy.  In  Fig.  S  two  circles,  with  their  involutes  satisfying  the 
above  condition,  are  drawn.  The  curves  may  be  regarded  as  the 
contours  of  a  right  circular  cone,  together  with  an  associated  surface 
which  is  formed  so  that  its  intersection  by  any  plane  parallel  to  that 
of  the  diagram  is  an  involute  of  the  circle  in  which  the  cone  is  cut 
by  the  same  plane. 

The  acceleration  and  speed  of  a  body  falling  under  the  action  of 
gravity,  and  the  space  passed  over  by  it,  are  given  by  the  known 
equations  (§  42) 

a  =  g 

v=  V+gt 

s  =  c  +  Vt  +  $fft*. 

Hence  these  various  quantities  can  be  represented  also  by  the 
contours  of  the  surfaces  just  considered,  £  'taking  the  place  of  0,  an# 
g  taking  that  of  a  in  the  former  equations.  The  new  terms  which 
.appear  present  no  difficulty.  •  • 

Again,  in  a  thermo-electric  circuit  (Chap.  XXVIII.),  composed  of 
two  dissimilar  metals,  the  electromotive  force  E  is  given,  in  terms 
of  t  the  difference  of  temperature  of  the  two  junctions,  by  means  of 
the  formula 


Also  the  thermo-electric  power  e  is  given  by  the  equation 


Hence  these  quantities  are  also  representable  by  means  of  the  same 
surfaces. 
23.  The  contour  lines  which  are  most  familiar  to  us  are  those 


THE   THEORY   OP   CONTOURS,   AND   ITS   PHYSICAL   APPLICATIONS.      25 

formed  by  the  intersection  of  level  surfaces  with  the  surface  of  the 
earth.  The  line  of  sea-board  is  one  such  contour  line.  The  numbers 
marked  upon  maps  or  charts  which  give  the  height  above,  or  depth 
below,  sea-level  indicate  contour-points.  When  the  points  are 
taken  sufficiently  close  together,  and  continuous  curves  are  drawn 
through  points  of  constant  height,  we  get  contours,  as  in  the 
Ordnance  Survey  maps.  Such  a  contour  coincides  very  closely  with 
the  contour-lines  formed  by  level  surfaces.  They  do  not  exactly 
coincide  because  of  the  non-spherical  shape  of  the  earth,  and  because 
of  its  rotation,  etc.  But  the  assumption  that  contour-lines  of  constant 
level  are  lines  of  constant  height  over  sea-level  will  not  introduce 
appreciable  error,  so  long  as  the  area  on  which  they  are  drawn  is 


FIG.  9. 

small  in  comparison  with  the  whole  surface  of  the  earth.  The 
kinetic  energy,  which  is  acquired  by  a  body  in  falling  freely  from  any 
point  on  one  level  line  to  any  point  on  another  level  line  is  constant. 
Suppose  the  earth  to  be  entirely  submerged  underneath  the 
surface  of  water,  so  that  we  have  only  one  region,  and  that  a  region 
of  depression  below  the  surface  of  the  water,  to  consider.  If  we 
suppose  further  that  the  water  is  slowly  absorbed  by  the  solid  matter 
of  the  earth,  regions  of  elevation  will  be  formed  gradually,  until 
finally  we  shall  have  again  only  one  region,  and  that  a  region  of 
elevation,  Before  a  region  of  elevation  is  formed,  we  have  a 
summit  appearing  above  the  water-level;  and  when  the  water 
subsides  out  of  a  region  of  depression,  we  have  a  lowest-point  or 
imii  appearing. 


26  A   MANUAL    OF   PHYSICS. 

The  number  of  regions  of  elevation  and  depression  may  vary  in 
two  ways.  Two  regions  of  elevation  may  run  into  each  other  as 
the  water  sinks.  The  point  where  they  first  meet  is  termed  a  jimss 
or  col  (see  Fig.  9  ;  Pj,  P2,  etc.).  Again,  a  region  of  elevation  may 
throw  out  arms  which  run  into  each  other,  and  so  cut  off  a  region 
of  depression.  The  point  where  they  first  meet  is  termed  a  bar 
(Blt  jB2,  etc.).  The  contour-line  for  a  level  immediately  underneath 
that  corresponding  to  the  bar  has  a  closed  branch  within  the  region 
of  depression  cut  off.  Thus  the  closed  curve  at  I4  is  part  of  the 
contour-line  UV.  In  the  map  of  such  a  country,  a  pass  occurs 
at  the  node  of  a  figure-of-eight  curve  (or  out-loop  curve,  as 
Professor  Cayley  has  termed  it),  while  a  bar  occurs  at  the  node 
of  an  in-loop  curve.  If,  in  the  diagram,  the  Ps  represented 
bars  and  the  Bs  represented  passes,  the  map  would  be  that  of  an 
inland  basin ;  so  that,  in  the  map  of  such  a  country,  a  pass  is  repre- 
sented by  the  node  of  an  in-loop  curve,  and  a  bar  corresponds  to 
the  node  of  an  out-loop  curve.  If  there  were  any  advantage  in 
having  passes  and  bars  always  indicated  by  the  node  of  the  same 
kind  of  curve  respectively,  this  could  be  attained  by  affixing  the 
positive  sign — not  constantly  to  the  region  on  the  same  side  of  the 
level  surface  but — to  the  region  towards  which  (or  from  which)  the 
surface  is  moving  at  any  instant. 

As  a  particular  case,  two  regions  of  elevation  may  run  into  each 
other  at  a  number  of  points  simultaneously.  Of  these  points,  one 
must  be  taken  as  a  pass  and  the  others  as  bars.  Singular  points 
may  also  occur,  when,  for  example,  three  or  more  regions  of  eleva- 
tion meet.  Such  points  are  called  double,  treble,  etc.,  passes. 
Multiple  bars  may  similarly  occur. 

Before  a  pass  can  be  formed  there  must  be  two  summits,  and  for 
every  additional  pass  there  is  another  summit.  Thus  the  number 
of  summits  is  one  more  than  the  number  of  passes.  So  also  the 
number  of  imits  is  one  more  than  the  number  of  bars. 

Slope-lines  are  lines  drawn  at  right  angles  to  the  contour-lines ; 
and,  evidently,  the  steepness  of  a  district  is  indicated  011  a  map  by 
the  closeness  of  the  contours.  Two  kinds  of  slope-lines  are  of 
special  importance.  These  are  the  slope-lines  drawn  from  summits 
to  passes  or  bars,  and  from  passes  or  bars  to  imits.  The  former  can 
never  reach  an  imit,  and  are  termed  water-slieds.  The  latter  can 
never  reach  a  summit,  and  are  called  water -courses. 

A  perpendicular  precipice  is  indicated  on  a  chart  by  the  running 
together  of  two  or  more  adjacent  contour-lines  (F).  An  over- 
hanging precipice  is  indicated  by  the  lapping  of  the  upper-level  line 
over  a  lower-level  line. 


THE   THEORY   OF   CONTOURS,   AND   ITS   PHYSICAL   APPLICATIONS.      27 

24.  Since  we  can  represent  the  physical  state  of  a  substance  with 
regard  to  three  quantities  by  means  of  a  surface,  it  follows  that  we 
can  deduce  from  the  contours  of  the  surface  the  nature  of  the  varia- 
tion of  the  properties  of  the  substance — the  methods  being  identical 
with  those  of  the  preceding  paragraph.  Let  us  take,  as  a  particular 
example,  the  thermo- dynamic  surface  which  represents  the  state 
of  water-substance  with  regard  to  volume,  pressure,  temperature, 
entropy,  and  energy  (Chap.  XXV.).  If  we  construct  the  surface 
in  terms  of  any  three  of  these  quantities,  the  value  of  the  remain- 
ing two  at  any  point  of  the  surface  may  be  given  by  contour-lines. 
The  model  of  the  surface,  having  volume,  entropy,  and  energy 
measured  along  the  axes,  has  been  constructed  by  Clerk  Maxwell, 
and  is  explained  and  figured  in  his  '  Theory  of  Heat.'  We  shall 


FIG.  10. 

consider  the  surface  representing  directly  volume,  temperature,  and 
pressure.  This  surface  was  first  studied  by  Professor  James  Thomson. 
Suppose  the  surface  to  be  cut  by  a  plane  of  constant  pressure,  say  Plf 
We  thus  get  a  contour-line,  the  general  nature  of  which  is  indicated 
in  Fig.  10.  At  a  low  temperature  the  volume  is  small,  the  sub- 
stance being  in  the  solid  state.  As  the  temperature  rises,  the  substance 
expands  until  liquefaction  occurs.  The  volume  then  diminishes 
without  rise  of  temperature,  until  the  substance  is  completely 
liquefied.  The  temperature  then  rises  while  the  volume  diminishes, 
until  the  maximum-density  noint  is  reached.  After  this,  expansion 
accompanies  rise  of  temperature  up  to  the  boiling-point.  At  this 
stage  the  volume  rapidly  increases,  while  the  temperature  remains 
steady  until  the  substance  is  entirely  in  the  gaseous  state.  Beyond 
this  point  both  increase  together.  The  contours  for  slightly  less 
pressures  (P2,  P3)  are  approximately  parallel  to  P15  but  lie  entirely 


A   MANtlAt   OF   PHYSICS. 

above  it ;  the  reason  being  that  at  a  given  temperature  the  volume 
increases  as  the  pressure  diminishes,  while  the  freezing-point  is 
lowered,  and  the  boiling-point  is  raised,  by  pressure.  The  freezing 
and  boiling  points  approach,  and  finally  coincide  as  the  pressure 
diminishes.  At  lower  pressures  the  substance  changes  directly  froni 
the  solid  into  the  gaseous  state.  The  line  AB  in  the  figure  above 
indicates  the  triple-point  temperature,  that  is,  the  temperature  at 
which  portions  of  the  substance  in  the  three  states — solid,  liquid, 
and  gaseous — can  exist  together  in  equilibrium.  The  increase  of 
volume  on  vaporisation  continually  diminishes  as  the  pressure  is 
raised,  until  finally  (at  .C)  the  process  of  vaporisation  ceases.  The 
temperature  at  which  this  occurs  is  called  the  critical  temperature'. 
There  may  also  be  a  critical  temperature  for  the  solid-liquid  con- 
dition. .  That  is  to  say,  there  may  be  a  temperature  belo^v  which 
no  amount  of  pressure  will  lower  the  freezing-point  sufficiently  to 
admit  of  liquefaction. 

The  contour-lines  which  are  obtained  by  cutting  the  surface  by 
planes  of  constant  temperature  are  called  isothermals.  If  the  tem- 
perature be  above  the  triple-point,  but  below  the  critical-point,  while 
the  substance  is  in  the  gaseous  condition,  increase  of  pressure  is 
accompanied  by  decrease  of  volume,  until  the  liquefaction  com- 
mences. At  this  stage  the  volume  decreases  without  variation  of 
pressure,  until  all  the  substance  is  liquefied.  After  this,  a  very 
great  increase  of  pressure  is  required  to  produce  a  very  small 
decrease  of  volume.  Two  such  isothermals  (not  those  of  water- 
substance,  see  §  278)  are  represented  in  Fig.  11.  The  form 
of  an  isothermal  below  the  triple -point  shows  that  the  solid 
state  is  intermediate  between  the  gaseous  and  the  liquid  states. 
As  the  pressure  increases  the  volume  decreases,  until  the  point 
of  sublimation  is  reached.  The  pressure  then  remains  constant 
while  the  volume  diminishes,  until  all  the  substance  is  solidified, 
Then  the  volume  decreases  slowly  on  increase  of  pressure,  until 
liquefaction  commences.  Here  the  pressure  becomes  constant, 
while  the  volume  diminishes  until  all  the  ice  is  melted ;  after 
which  the  volume  again  decreases  slowly  on  rise  of  pressure. 
Thus,  here  are  two  kinds  of  isothermals  having  their  transition- 
stage  at  the  triple-point  temperature.  As  already  indicated,  the 
triple-point  pressure  occurs  at  the  transition  between  two  kinds  of 
lines  of  equal  pressure,  for  the  liquid  condition  ceases  to  be  possible 
at  higher  pressures.  The  form  of  the  isothermals  beyond  the 
critical  temperature  is  indicated  in  Fig.  11.  It  is  quite  possible 
that,  as  Professor  James  Thomson  has  suggested,  the  true  form  of 
the  isothermals  below  the  critical  temperature  does  not  include  a 


THE   THEORY   OF   CONTOURS,   AND   ITS   PHYSICAL   APPLICATIONS.        29 


FIG.    11. 


30  A   MANUAL    OF    PHYSICS. 

part  parallel  to  the  volume-axis,  but  that  it  has  a  waved  form,  as 
shown  in  the  diagram.  Part  of  the  waved  portion  represents  an 
unstable  state,  since  pressure  and  volume  increase  together.  Thom- 
son made  this  suggestion  in  order  to  avoid  discontinuity  in  the  curve. 

The  lines  elt  e%  are  lines  of  constant  energy,  and  those  marked 
0i »  02 »  03  are  lines  of  constant  entropy. 

25.  It  is  only  when  the  temperature  considered  happens  to  be 
one  corresponding  to  a  contour  in  the  diagram  that  the  relation  of 
pressure  and  volume  can  be  accurately  found  from  the  above 
diagram.  This  defect  may  be  got  rid  of  by  the  use  of  trilinear 
co-ordinates;  and,  in  addition,  the  variation  of  a  fourth  quantity 
can  be  shown.  In  illustration  of  this  we  may  take  the  case  of  a 
perfect  gas,  for  which  we  have  the  equation 

vv  =  lit, 

where  p,  v,  and  t  represent  respectively  the  pressure,  volume,  and 


FIG.  12. 

temperature,  and  R  is  a  quantity  which  depends  on  the  nature  of 
the  gas.  The  triangle  of  reference  is  made  equilateral  in  Fig.  12. 
The  ratios  of  the  distances  of  a  point  from  the  vertical  and  the 
inclined  sides  of  the  triangle  are  the  ratios  of  the  temperature, 
pressure,  and  volume  respectively.  The  contours  for  different  values 
of  R  are  shown  in  the  figure,  and  the  equation  shows  that  they  are 
hyperbolas,  with  vertical  and  horizontal  axes.  No  part  of  the 
hyperbolas  outside  the  triangle  of  reference  has  any  physical  mean- 
ing; for,  in  that  case,  pressure,  volume,  or  temperature  (or  any 
two,  or  all,  of  them)  would  be  negative,  which  is  impossible  in  the 
case  of  a  gas.  Evidently  pressure,  volume,  and  temperature  are 
continuously  represented  for  any  one  gas. 


THE    THEORY    OF   CONTOURS,    AND    ITS    PHYSICAL   APPLICATIONS.      81 

Of  course,  when  we  wish  to  find  the  absolute  values  of  the  co- 
ordinate quantities,  we  must  use  the  equation  which  connects  them. 
This  is  not  necessary  when  we  use  Cartesian  co-ordinates. 

The  figure  obviously  represents  the  contours  of  a  surface  by  planes 
parallel  to  the  plane  of  the  diagram,  which  may  be  looked  upon  as 
that  corresponding  to  zero  value  of  R.  When  li  is  zero,  the 
hyperbola  becomes  a  pair  of  straight  lines  which  coincide  with  the 
sides  AB,  AC,  of  the  triangle  of  reference.  When  E  is  infinite,  the 
side  BC  is  part  of  the  corresponding  hyperbola.  All  parts  of  lines 
through  B  and  C  perpendicular  to  the  plane  of  the  diagram  lie 
upon  the  surface.  These  lines  separate  the  parts  of  the  surface 
which  correspond  to  real  physical  states  from  those  which  do  not. 
Outside  the  triangle  the  surface  evidently  overhangs  the  plane  of 
the  paper. 

The  value  of  R  being  given,  let  P  (Fig.  18)  be  the  point  which  gives 


the  proper  ratios  of  p,  v,  and  t.  Draw  PM ,  PN,  parallel  to  the  sides 
of  the  triangle.  Since  the  asymptotes  of  the  hyperbola  are  parallel 
to  these  sides,  it  follows  that  that  part  of  the  tangent  at  P,  which  is 
intercepted  by  the  sides  of  the  triangle,  is  bisected  at  the  point  of 
contact.  Therefore  AM  =  MQ,  and  AN  =  NR.  Now  the  compres- 
sibility k  of  a  gas  is  given  by  the  ratio  dv/vdp,  where  dv  is  the 
small  alteration  of  volume  produced  by  the  small  change  of  pressure 
dp.  Vutdvldp  =  MQIMP  =  NPIMP  =  vlp.  Hence  k  =  1/p,  that  is, 
the  compressibility  of  a  perfect  gas  is  the  reciprocal  of  the  pressure. 
Similarly  it  can  be  shown  that  the  expansibility  is  proportional  to 
the  absolute  temperature. 

The  work  done  during  isothermal  expansion  can  also  be  found 
from  the  diagram.     The  position  of  the  point  P  gives  the  mutual 


32  A   MANUAL   OF   PHYSICS. 

ratios  of  p,  v,  and  t ;  but  since  t  has  a  known  constant  value,  the 
actual  values  of  p  and  v  are  also  known.  Hence  PN  ( =p  cosec 
BAG)  is  a  known  function  of  v.  If  P  move  to  P',  the  area 
PNN'P'  =/PNdv  (§  34)  =  cosec.  BACfpdv  is  a  known  multiple 
of  the  work  done. 

;  26.  The  applicability  of,  the  method  of  contours  to  other  physical 
problems  is  evident.  Electric  stream-lines  and  equipotential-lines 
may  be  regarded  as  contours  of  a  surface,  and  the  number  of  equi- 
potential  lines  which  cross  unit  length  of  a  stream-line  may  be  used 
to  indicate  the  strength  of  the  current.  So  also  air- current  lines  and 
isobars,  isothermals  and  flow-lines  of  heat,  etc.,  are  rectangular 
systems  of  contours. 


CHAPTEE  IV. 

VARYING   QUANTITIES. 

27.  THOUGH  every  quantity,  whatever  be  its  nature,  has  magnitude, 
no  quantity  can  be  said  to  be  large  or  small  absolutely.  When  we 
speak  of  the  size  of  any  body  we  mean  its  size  relatively  to  the  size 
of  some  other  body  with  which  we  compare  it.  A  yard  is  large 
if  we  compare  it  with  an  inch ;  it  is  small  when  compared  with  a 
mile.  In  the  former  case  the  number  which  represents  it  is  more 
than  60,000  times  larger  than  the  number  by  which  it  is  represented 
in  the  latter  case.  A  mere  number  is  therefore  useless  as  regards 
the  statement  of  magnitude,  except  when  accompanied  by  a  clear 
indication  of  what  the  thing  measured  is  compared  with.  The 
quantity  in  terms  of  which  the  comparison  is  made  is  called  the 
unit,  and  the  number  which  tells  how  often  this  unit  is  contained 
in  a  given  quantity  is  called  the  numeric. 

All  dynamical  quantities  may  be  made  to  depend  upon  three  units 
only.  These  are  the  units  of  mass  (quantity  of  matter),  length, 
and  time.  Thus  speed,  being  measured  by  the  distance  traversed 
in  a  certain  time,  depends  upon  the  unit  of  length  directly, 
and  upon  the  unit  of  time  inversely.  Hence  by  doubling  the 
unit  of  length  we  double  the  speed  unit,  and  therefore  halve  the 
numeric  of  any  given  speed ;  whereas  by  doubling  the  unit  of  time 
we  halve  the  speed  unit,  and  therefore  double  the  numeric  of  a 
given  speed.  Again,  acceleration,  being  measured  by  the  increase 
of  speed  in  a  certain  time,  depends  upon  the  unit  of  speed  directly 
and  upon  the  unit  of  time  inversely  ;  that  is,  it  depends  directly 
upon  the  unit  of  length  and  inversely  upon  the  square  of  the 
unit  of  time.  The  manner  in  which  the  fundamental  units 
are  involved  in  any  quantity  determines  the  dimensions  of  that 
quantity.  If  M  L  and  T  represent  the  units  of  mass,  length,  and 
time,  the  dimensions  of  speed  and  acceleration  are  indicated  by 
the  symbols  [LT"1]  and  [L27-2]  respectively,  and  the  dimensions 
of  energy  (§  7)  by  [ML2T-2]. 

3 


34  A   MANUAL    OF    PHYSICS'. 

28,  When  two  quantities  are  so  related  that  any  change  in  the 
numerical  value  of  one  of  them  is  accompanied  by  a  change  in  the 
numerical  value  of  the  other,  each  quantity  is  called  a  function  of 
the  other.     If  to  one  value  of  the  first  there  corresponds  one,  and 
only  one,  value  of  the  second,  the  second  is  called  a  single-valued 
function  of  the  first ;    but  if  to  a  given  value  of  the  one  there 
correspond  more  than  one  value  of   the  other,  the  latter  is  said 
to  be   a  multiple-valued  function  of   the  former.     For  example, 
a;24-2a?-8  is  a  single-valued  function  of  x;  since,  if  we  give  x  any 
value,  there  will  be  one  corresponding  value  of  a;2-f  2#  —  8.     On  the 
other  hand,  a?  is  a  double-valued  function  of  #2+2a?  —  8;  since,  if 
we  give  any  value  to  #2-f  2#  —  8,  we  find  that  x  has  in  general  two 
distinct  values.     Again  sin  a?  is  a  single-valued  function  of  x,  while 
sin"1^  (the  angle  whose  sine  is  x}  is  a  multiple -valued  function  of  #, 

The  relation  between  the  two  quantities  can  be  expressed  by 
means  of  an  analytical  equation  or  by  means  of  a  curve,  as  is  indi- 
cated in  §  17.  The  general  expression  of  the  relation  may  be  given 
in  the  form 

0  =/(»). 

where  the  quantity  y  is  regarded  as  being  dependent  upon  x,  and 
the  equation  simply  reads  '  y  is  some  function  of  #.'  The  function 
denoted  by  /  may  be  of  various  kinds — it  may  be  algebraical 
(e.g.,  y  =  ax  +  bx2),  trigonometrical  (e.g.,  y  =  sin  a?-fcos  x),  ex- 
ponential (e.g.,  y  =  a*),  etc. ;  and  under  each  kind  there  may  be 
a  number  of  different  forms,  e.g.,  a  'series  of  powers,'  a  'product 
of  tangents,'  etc. 

In  the  above  formula  x  is  supposed  to  be  that  quantity  the  value 
of  which  is  arbitrarily  varied.  It  is  therefore  called  the  independent 
variable,  while  y  is  termed  the  dependent  variable. 

29.  If  y  varies  in  value  uniformly  when  x  varies  uniformly,  the 
quotient  of  any  increment  of  y  by  the  simultaneous  increment  of  x 
is  constant,  however  large  or  small  either  of  the  increments  may 
be.     For  the  condition  means  that  y  is  proportional  to  x,  or  to  x 
+  a  constant,  say  y=~kx-\-  c  where  k  and  c  are  constants.     Hence 
if  y  changes  from  y1  to  y2  when  x  changes  in  value  from  xl  to  x.2 
we  have  yl  =  kxi-}-c,  y2=kx2+c,  and  therefore  y2-yi  =  k  (x.2  —  x^, 
which  proves  the  above  statement,  since  a?2  —  x\  may  have  any  value 
we  please.     If  the  values  of  y  and  x  be  represented  as  the  ordinates 
and  abscissae  respectively  of  a  curve,  we  see  that  the  constant  c  is 
represented  by  the  length  OA  (Fig.  14)  (since  it  is  the  value  of  y  when 
x  =  o),  that  simultaneous  values  of  x  and  y  are  represented  by  points 
on  the  line  AB,  and  that  k  is  represented  by  the  tangent  of  the  in- 
clination of  that  line  to  the  #-axis  ;  hence,  by  this  graphical  process 


VARYING    QUANTITIES. 


3,5 


also,  the  truth  of  the  statement  is  evident.     When  they  are  thus 
related,  y  and  x  are  said  to  be  linear  functions  of  each  other. 

But  when  y  is  not  a  linear  function  of  x,  it  is  evident  that  the 
ratio  of  their  simultaneous  increments  depends  upon  the  absolute 
values  of  these  increments.  In  the  former  case  that  ratio  denoted 
the  rate  at  which  y  varied  when  x  varied  ;  but,  in  the  case  now 
supposed,  it  does  not  in  general  give  the  true  value  of  that  rate. 
It  is  of  extreme  importance  that  we  should  have  a  means  of  finding 
the  true  rate  of  variation  whatever  be  the  nature  of  the  relation 
connecting  the  two  quantities.  Let  A'B'  (Fig.  14)  be  the  curve 
which  represents  y  as  a  function  of  x,  and  let  P  be  the  point  at 
which  we  wish  to  find  the  rate  of  variation  of  y  when  x  alters. 
Take  another  point  P'  on  the  curve,  and  let  x'  and  y'  be  the  values 
of  its  co-ordinates.  It  is  obvious  that  in  general  the  ratio  of  y'  —  y 
to  x'  —  x  does  not  give  the  true  rate.  It  gives  instead  the  rate  of 


FIG.  14. 

variation  corresponding  to  the  line  joining  P  and  P'.  But  the  line 
PP'  coincides  more  and  more  nearly  with  the  curve  as  P'  approaches 
to  P;  and  we  can  take  P/  closer  to  P  than  any  assigned  finite 
quantity  however  small,  so  that  the  difference  in  direction  between 
the  line  and  the  curve  can  be  made  smaller  than  any  assignable 
angle.  Ultimately — when  P'  is  infinitely  near  to  P — the  rates  of 
variation  corresponding  to  the  curve  and  the  line  are  identical. 
Hence,  geometrically,  the  true  rate  at  any  point  is  given  by  the 
tangent  of  the  angle  which  the  line  touching  the  curve  at  that  point 
makes  with  the  #-axis ;  or,  analytically,  it  is  given  by  the  ratio 
°f  y'  ~  y  to  x'  —  x  when  x'  -  x  is  made  indefinitely  small. 

This  method  is  only  applicable  strictly  to  the  case  of  quantities 
which  have  no  sudden  change  in  their  rate  of  variation.     But,  in 

3—2 


86  A  MANUAL   OF   PHYSUS. 

any  such  case,  we  can  apply  the  method  up  to  the  point  at  which 
the  sudden  change  occurs,  and  also,  separately,  beyond  this  point : 
and  this  is  all  that  is  required. 

It  is  usual  to  write  dy  and  dx  instead  of  y'  —  y  and  x'  —  x  when 
these  quantities  are  infinitely  small,  so  that  the  symbol  d  means 
the  infinitely  small  increment  (or  *  differential,'  as  it  is  usually 
termed)  of  the  quantity  to  which  it  is  prefixed.  In  general.  <///  d ,r 
is  a  function  of  x  ;  so  that,  if  the  original  relation  is  y=f(jc\  it  is 
usual  to  represent  the  quantity  dyldf  by  the  symbol  f\x).  and  to 
call  it  the  first  derived  function  of  y  with  respect  to  x.  We  may 
deal  similarly  with  the  relation  y—f'(x),  and  obtain  the  second 
derived  function  which  is  indicated  by /"(a:),  and  so  on.  It  must 
be  carefully  observed  that  dx  and  dy,  being  quantities  of  the  same 
kind  as  x  and  //,  are  subject  to  identically  the  same  laws. 

30.  We  shall  now  find  the  rates  of  variation  of  .certain  func- 
tions which  will  be  of  use  subsequently;  first  of  all,  of  rational 
a  /</«'?)  raical  functions. 

a  and  6  being  constants.     We  shall  use  the  ordinary  sign  for  a 

limit,  according  to  which  J^j  denotes  the  limiting  value  of  the 
T*V  *•»• 

quantity  to  which  the  sign  is  prefixed  when  in  that  quantity  we 
put  x'—xt  i.e.)  make  x'-x  infinitely  small.  We  have 

^  =  JjZ^x==L  (x'^W=a' 
x'—x  x'=z 

a  result  which  has  already  been  given  in  the  preceding  section. 

This  example  shows  that  a  constant  term  in  a  function  does  not 
appear  in  its  derived  function. 

(2.)  y  =  ax* 
This  gives 

dy_J  a(x>*-x*) _ ^a(x>+x)(x'-x)  =  ^ ^ 

»'=»    '  •    • 

(8)  y  =  ax», 


f-x 

.'-j.)  (x'H-1+am~9g+  .  .  . 


X  -X 
**— X 


VARYING   QUANTITIES.  37 


— 

(4.)  y  —  axn. 
This  gives 

//"  r=  fall    Jj"1. 

d(yn)  _ 
dx  dx 

<%*)     dy  _ 
dy      dx  dx 

ny*-l?y=a*mx™-i     ..........     ty  (3) 


xm-l 


m     xm~1      in  „,_,_„,.»*       m  - — i 

=  a -,  =  a—xm  n  =a—x*     , 

n       m«zl       n  n 


We  see  from  (3)  and  (4)  that  when  y  is  given  as  a  positive  power  of 
x  (whole  or  fractional),  its  #-rate  of  variation  is  got  by  the  following 
rule  :  Multiply  by  the  index  of  x  and  lower  the  index  by  unity. 
Example  (7)  will  show  that  the  rule  is  not  restricted  to  positive 
powers  of  x,  but  holds  also  for  negative  powers. 

(5.)  y  =  axn-{-bxm 

djtj  r<t(»7-»H)+i(^-»ro)-|=T  r«(^-i+8.-«x+  .  0 

dx    JJL     x-x  x'—x     J     LJL 

x'=x  x'=x 


=  anxn~l  -\-  bmxm~l  '. 

From  this  example  we  see  that  the  rate  of  variation  of  a  sum  of 
such  terms  'is  the  sum  of  the  rates  of  variation  of  each. 

(6.)  y  =  wo, 
where  u  and  v  are  functions  of  x.    As  before 


_ 

dx 

x'=x 


and,  just  as  y  increases  by  the  quantity  dy  when  x  increases  by  dx, 
u  and  v  will  increase  by  du  and  dv.    Therefore 

dy     (u+du)  (v-\-dv)-uv    udv+vdu+dudv 
dx~  dx  dx 


38  .      A  MANUAL   OF   PHYSICS. 

But  the  third  term  in  the  numerator  of  this  fraction,  being  the 
product  of  infinitesimally  small  quantities,  is  infinitely  smaller 
than  the  others,  and  so  vanishes  in  comparison  with  them. 
Hence 

dy  =  d(uv)  =  udv + ^du^ 

dx       dx         dx      dx 

That  is  to  say,  the  rate  of  variation  of  a  product  of  two  quantities 
is  obtained  by  taking  the  sum  of  the  products  of  each  quantity 
into  the  rate  of  variation  of  the  other.  As  an  illustration  we  may 
find  the  #-rate  of  variation  of  xm  .  xn.  Here  the  result  is 

xm  .  nxfi-l+m&n-1 .  xn 

which  agrees  with  the  result  of  (3). 

If,  in  the  above  example,  y  is  a  constant,  i.e.,  if  the  product  of 
u  and  v  is  constant,  we  get  udv  =  -  vdu. 

(7.)  ,-*-.  ' 

Let  v  =  ~  and  therefore  y  =  uw,  and  we  get 


dx         dx         dx 

But,  as  has  just  been  remarked,  we  have  vdw  =  -  ivdv,  since  wv  =  l, 
and  so 

dy  _     du    uw  dv  _  1  du     u     dv 
dx         dx      v    dx~~ v  dx     v'2  '  dx 
du        dv 
dx         dx 

~^2  > 

which  gives  the  rule  for  the  rate  of  variation  of  a  quotient  of  two 
functions. 

If  u  =  xn,  v  =  xm,  we  get 
dy  _xn 


dx  x*™ 

Again,  if  u  is  constant,  we  have 

dx        -y2    dx' 

Thus,  if  2/  =  aar»,  a  takes  the  place  of  u,  and  xP  replaces  v.     There- 
fore 

dx         x^n     dx         x^n 


VARYING    QUANTITIES.  39 

which  proves  the  applicability,  to  the  case  of  negative  indices,  of  the 
rule  given  under  example  (4). 

In  many  cases  y  will  be  given,  not  in  terms  of  x  as  hitherto 
assumed,  but  in  terms  of  some  function  of  x,  say  u.  The  ordinary 
principles  of  algebra  then  show  (see  the  remark  at  the  end  of  last 
section)  that 

dy  _  dy    du 

dx     du    dx' 


For  example,  y  =  ( 

dy  _  d(ur]  _  ckt*  du 
dx       dx       du   dx 

te-iffc-^-^+^-i,-!^ 

dx  dx  dx 


31.  We  shall  now  consider  such  rates  of  variation   of  trigono- 
metrical functions  as  may  be  of  use  later. 

(1)  2/  =  sina?. 

di/T  sin  a;'  -sin  a;         T   sin  (f  ~  I)  cos  (f  +  I  ) 

dx     LJ         x'  —  x  JU  x'  —  x 

x'=x  x'=x 

But,  when  x'  =  x,  sin  (xf  -  x)  =  x'  -  x,  and  therefore  the  rate  of 
variation  is  cos  x. 

It  follows  at  once  from  this  result,  by  writing  #  +  o  instead  of  #, 

that  if 

(2)  2/  =  cos  x 

^=-sin*. 
dx 
Again,  suppose 

(3)  7/  =  sec  x. 

Let  cos  x  =  u,  and  we  get 


^ 

dx     du    dx  dx  '   dx 

=  sec2  x  .  sin  #  =  sec  x  tan  x. 

From  this  result  we  readily  obtain,  when 
(4)  ?/  =  cosec  x, 

-^-=  -cosec  x  cot  x. 
dx 


40 


A    MANUAL    OF    PHYSICS. 


By  means  of  the  result  of  example  (7)  in  §  30  we  can  find  the 
derived  function  of 

sin  x 


(5)  y  =  tan  x 


That  result  gives 


cos  x 


ax 


dx 


COS'  X 


_cos2  x-\-sm2x_ 

COS2  X  COS*  X 

and,  as  formerly,  by  the  substitution  of  x-\--  for  x,  we  deduce,  from 

(6)  y  =  cot  x, 
_=-cosec2z. 

32.  It  will  be  instructive  to  deduce  the  results,  which  we  have 
just  obtained,  by  a  geometrical  process.  Take  two  near  points, 
P  and  P',  on  the  circumference  of  the  circle  APB.  Denote  POA 
by  9  and  POP'  by  d9,  and  suppose  that  the  radius  of  the  circle  is 


FIG.  15. 

unity.  Let  OM  =  x  and  MP  =  ?/;  then  NP'  =  cfo/  and  -NP  =  d.c. 
The  angle  d9  is  assumed  to  be  infinitesimally  small  (so  that  the  arc 
PP'  is  practically  a  straight  line),  and  it  is  measured  by  the  ratio  of 
PP'  to  OP;  that  is,  by  PP',  since  OP  is  unity.  Also  NP'P  =  0. 
Hence  NP'/PP' = cos  9  =  dy[dO,  and  -  NP/PP'  =  -  sin  9  =  dx/dO.  But 
x  =  cos  0,  y  =  sin  9 ;  and  therefore 


Also 

and  therefore 


^H^  =  cos  0,  ^^  = 
d9  d9 

sec  9  =  !/«£, 


-  sin  9. 


d  sec  9  = 


-dx 


x  +  dx      x     x(x  -f  dx) 


dx 

X* 


VARYING   QUANTITIES.  41 

ultimately.     But 

-^  =  1  (  -^}d6  =  i  sin0  dO  =  sec8  0  sinO  d9  =  seceta,nOdO. 
ar       x2  \      dO/  x~ 

Similarly, 


d  tan  9  =  y^       -U 


+  dx      x  x(x-\-  dx) 


cos  0  .  cos  9  d  9  -  sin  9  (  -  sin  0)  d  9      ^ 
=  —  —  2-0  —  s  —  —  ^(l+tan2^)  d0  =  sec20  d9. 

COS    v 

33.  Lastly  we  may  obtain  the  rate  of  variation  of  the  exponential 
and  logarithm  of  a  varying  quantity. 

(1)  y  =  ex 


dj/  =  T   &f-e?=  T 
^a?~  LJ  x'-x       JLj 


x'-x 

x'=x 


But  e,  which  is  the  base  of  the  Napierian  Logarithms,  is  by  defini- 
tion the  limiting  value  of  (1  +  u)u  when  u  is  a  vanishingly  small 
quantity  ;  so  that  we  may  write 


1 

X'=X 

Therefore  the  limiting  value  of  ex'~x  is 

x'-x 


J        fe 


a;—  a; 


and  therefore 


By  means  of  this  result  we  can  deduce  the  derived  function  of 

(2)   y  =  log  x. 

For  this  gives  x  —  ey,  and  therefore  dx/dy  =  ey,  so  that 


dx 

Inverse  Problem. 

34.  In  the  immediately  preceding  sections  we  have  dealt  with 
the  problem  —  to  find  the  rate  of  variation  of  a  quantity  which  is  a 
given  function  of  some  independent  variable.  The  inverse  pro- 
blem —  to  find  the  value  of  a  quantity,  the  rate  of  variation  of  which 
is  given  —  is  of  quite  as  great  importance  in  physical  inquiries  ;  and, 
in  order  to  solve  it,  we  have  merely  to  reverse  the  former  process. 


42 


A    MANUAL    OF    PHYSICS. 


In  illustration  we  may  consider  the  simplest  possible  case  in  which 
we  have  y  =  x.  Here  dyjdx  is  constant  and  numerically  equal  to 
unity  (§  30,  example  (1) ).  The  relation  y^x  is  indicated  in  Fig.  16 
by  the  line  bisecting  the  angle  xoy,  and  the  values  of  dyjdx  are 


given  by  points  on  the  horizontal  straight  line  passing  through  the 
point  y  =  l.  If  we  choose  any  point  P  on  the  line  y  =  x,  and  draw 
through  it  a  line  parallel  to  the  2/-axis  cutting  the  line  indicating  the 
value  of  dyjdx  in  P',  it  is  evident  from  the  figure  that  the  area  of  the 


FIG.  17. 

rectangle  OP'  is  numerically  equal  to  the  value  of  y  at  P.  For  the 
ordinate  of  P  is  four  units  in  length,  and  OP'  contains  four  square 
units. 

Now  take  any  curve  (the  particular  curve  shown  in  Fig.  17  is  the 


VARYING    QUANTITIES.  43 

quadrant  of  a  circle)  representing  graphically  the  relation  ?/=/(#), 
and  draw  another  curve  representing  the  relation  y'=f'(x).  This 
may  be  called  the  '  derived '  curve  of  the  former,  since  its  ordinates 
give  the  values  of  dy/dx  at  the  corresponding  points,  i.e.,  they  give 
the  values  of  the  derived  function.  Take  three  near  points,  P^P^P.^ 
on  the  derived  curve.  The  area  included  between  the  curve  and  the 
axes  is  greater  than  the  sum  of  such  areas  as  P2#i,  but  is  less  than 
the  sum  of  such  areas  as  PiX2.  These  sums  differ  by  the  sum  of 
the  rectangles  PiP2,  etc.  But,  as  #2  —  x^  etc.,  are  made  less  and 
less  and  ultimately  vanish,  these  rectangles  become  smaller  and 
smaller  and  finally  vanish.  Of  course  any  area  PiX2  becomes 
infinitesimally  small  as  a?2  approximates  to  coincidence  with  x± ;  but 
PiP2  becomes  infinitely  small  in  comparison  with  P^ — infinitely 
small  though  it  be.  The  magnitude  of  each  little  area  similar 
to  Pj#2  is  y'dx,  where  dx  represents  as  before  an  infinitesimally 
small  increment  of  x.  But  y'— dy/dx,  and  therefore  y'dx  =  dy.  So 
that,  finally,  the  limiting  value  of  the  sum  of  the  rectangles  (the 
number  of  which  increases  indefinitely  as  their  magnitude  dimi- 
nishes without  limit)  up  to  a  given  value  of  x,  is  numerically  equal 
to  the  value  of  y,  at  that  given  value  of  a?,  in  the  original  curve. 
This  limit  to  which  the  sum  approaches  is  indicated  by  the  symbol 


fy'dx, 


and  the  quantity  itself  is  called  the  integral  of  y'  with  respect  to  x. 
That  is  to  say,  while  y'  is  called  the  '  derived  function '  of  y  with 
respect  to  x,  y  is  said  to  be  the  '  integral '  of  y'  with  respect  to  the 
same  variable  quantity.  If  it  were  necessary  to  preserve  similarity 
of  nomenclature,  we  might  term  y  the  '  primitive  function '  of  y'. 
Analytically,  the  operation  indicated  by  f  is  an  operation  which 
undoes  the  effect  of  the  operation  indicated  by  the  symbol  d.  For 


dx 

The  symbol  d  signifies  an  infinitesimal  difference ;  the  symbol  / 
signifies  the  sum  of  an  infinite  number  of  infinitesimally  small 
differences.  Indeed  the  symbol  of  integration  is  merely  an  exag- 
gerated form  of  the  letter  S,  denoting  a  sum. 

In  order  to  find  the  integral  of  y'  we  have  to  answer  the  question, 
What  function  has  y'  as  its  derived  function  ?  A  considerable  know- 
ledge of  derived  functions  is  therefore  essential. 

35.  We  shall  now  consider  some  useful  examples,  and  shall  take 
them  in  an  order  similar  to  that  which  was  observed  when  finding 
the  derived  functions. 


44  A   MANUAL   OF   PHYSICS. 

(1)  Evaluate  y=fadx. 

We  have  merely  to  write  down  that  function  the  derived  function 
of  which  with  respect  to  x  is  a.  In  the  direct  process  we  had  to 
lower  the  power  of  x  by  unity  and  multiply  by  the  undiminished, 
power  as  a  factor.  Therefore,  in  the  inverse  process,  we  must  raise 
the  power  by  unity  and  divide  by  the  increased  power.  But,  in 
the  above  example,  we  may  assume  x°  as  a  factor  since  its  value  is 
unity  ;  so  that  ax°  is  the  quantity  of  which  we  have  to  obtain  the 
integral.  In  accordance  with  the  rule  just  given  the  result  is  ax. 
But  we  must  remember  that  in  the  direct  process  all  constant 
terms  disappear  (§  30,  example  (1)  )  ;  so,  to  the  result  as  above 
obtained,  we  must  always  add  on  a  constant.  Thus  : 

y  =  J  adx  =  ax-\-b. 

The  constant  b  is  quite  arbitrary  unless  some  condition  is  laid  down 
which  determines  it.  Thus  the  condition  might  be  that  when  x  is 
unity,  y  is  equal  to  a  +  3,  in  which  case  we  see  that  the  value  of  6  is  3. 

(2)  Evaluate  y  =  Jlaxdx. 

In  accordance  with  the  rule  which  requires  us  to  increase  the  power 
of  x  by  unity  and  divide  by  the  power  so  increased  and  add  a  con- 
stant, we  get 

y  =  2a£z2  +  b  =  ax2  -f  b. 
Similarly  from 

(3)  y'  =  anxn~l 
we  deduce 

y  =  axn+b. 
Or,  changing  n  into  n  +  l»  from 

y'  =  axn 
we  obtain 


which  is  true  whether  n  be  positive  or  negative,  whole  or  fractional. 
The  proof,  given  in  §  30,  that  the  rate  of  variation  of  a  sum  of 
powers  of  the  independently  varying  quantity  is  the  sum  of  the  rates 
of  variation  of  each  term,  gives  at  once,  by  inversion,  the  rule  that 
the  integral  of  a  sum  of  powers  is  the  sum  of  the  integrals  of  each 
term.  From 

(4)  y'= 
we  can  at  once  write  down 

xn+1  ,  7 
v  =  a  -  +  b 

n+1         m+1 

In  example  (6),  §  30,  it  has  been  shown  that,  when  y  =  uv,  the 


VARYING    QUANTITIES. 


45 


increment  of  y  consequent  upon  alteration  of  the  quantity  of  which 
u  and  v  are  functions  is 

dy 
Hence,  from 

(5)  y'  = 
we  obtain 

y  —  uv  +  a  constant. 

Also  from  example  (7)  of  the  same  paragraph  we  see  that 


gves 


y  = 


u 

—  +  a  constant. 


36.  In  section  34  we  found  how  we  might  represent  graphically 
the  value  of  y  for  different  values  of  x  by  means  of  an  area  included 
between  the  a?-axis,  the  curve  representing  y'  as  a  function  of  a?,  and 
two  ordinates  of  that  curve.  By  means  of  this  method  we  can 
obtain  an  independent  proof  of  the  result  of  example  (3)  above. 
Let  the  curve  in  the  diagram  represent  the  relation  y'  =  xn.  The 
value  of  y  corresponding  to  x  is  represented  by  the  area  OxP  ;  and 


FlG.    18. 

OxP=OxPy'—OPy'.  Now,  in  precisely  the  same  way  that  OxP 
represents  fy'dx,  'the  area  OPy'  represents  the  quantity  fxdy',  as 
is  indicated  by  the  horizontal  rectangle.  But  fxdy'—fxd  xn  = 

=fx.  nxn~l .  dx  =  nfxndx.   Also  OxP  —fy'dx  =fxndx, 

and  OxPy'  =  xyf  =  xn+1.  Therefore  xn+l=fxndx  +  nfxndx  =  (n  + 1) 
fxndx  -=  (n  +  l)fy'dx  =  (n  +  l)y  ;  that  is 


/»•£ 


46 


A    MANUAL    OF    PHYSICS. 


In  Fig.  19  the  rectangular  area  OP  represents  the  product  of 
u  and  v.  "When  u  and  v  increase  or  decrease  simultaneously,  the 
increase  of  uv  is  evidently  udv  -f  vdu  (the  little  rectangle  at  P  being 
neglected).  And  also  uv  =fvdu  +/udv,  which  makes  the  result 
of  example  (5),  §  35,  almost  self-evident.  If  u  decreases  as  v  increases 
(as  in  passing  from  P/1  to  P'2)  the  area  uv  increases  by  the  quantity 
udv,  but  decreases  by  the  amount  vdu.  But,  in  this  case,  dit,' is 
itself  negative ;  so  that  the  result  is  still  d(uv)  =  udv  +  vdu. 


M 


M  u 

FIG.  19. 

If  the  curve  P  represents  the  value  of  v  in  terms  of  u,  we  may 
take  its  reciprocal  curve  P'  which  represents  w  in  terms  of  u,  where 

w  =  -•    The  diagram  gives  (as  above) 

uw  —  fudw  -\-fwdu. 
But  when  any  quantity  v  increases  by  the  amount  dv,  its  reciprocal 


decreases  by      - 


=  *  ultimately;  that  is,  the 


increase  is  -  — • 

v2 

Therefore 


u  _  fdu  _  fudv  _  f\ 
~  -J-j      J  -tfT  -J 


which  is  the  result  of  example  (6)  preceding. 

37.  From  the  results  of  examples  (1)  and  (2)  of  §  31,  we  can  at 
once  write  down 

/sin  xdx  =  —  cos  x  and  /cos  xdx  =  sin  x, 

because  the  derived  functions  of  cos  x  and  sin  x  are  respectively 
—  sin  x  and  cos  x. 


VARYING    QUANTITIES.  47 

Since  we  know  that  d  tan  9jd  0  =  sec2  0,  we  can  put 

tan0=/sec20d0; 
and  similarly 

/sec  0  tan  0  d  0  =  /sec2  0  sin  0  d  0  =  sec  0 

/cosec  0  cot  0  d  9=  /cosec2  0  cos  0  d  9—  -  cosec  0, 
and 

/cosec2  0  d0=  -cot  0. 

38.  Lastly,  since  (§  33)  dexldx  =  ex  and  d  log  xldx  =  ->  we  get 

0 

'  =  10£  X. 


CHAPTER   V. 

MOTION. 

39.  Position.  —  The  position  of  a  point  in  space  is  completely 
determined  when  three  independent  conditions  are  given,  each 
of  which  it  satisfies.  And  its  position  can  only  be  given 
relatively  to  that  of  another  point,  for  we  do  not  know  any 
point  of  which  we  can  assert  that  it  is  absolutely  fixed.  We 
may  say  that  one  point  is  so  much  to  the  north  or  south 
of  another,  so  far  to  the  east  or  west  of  it,  and  so  much 
higher  or  lower ;  or  we  may  say  that  it  is  so  far  distant  from  the 
other,  that  the  line  joining  the  two  is  inclined  at  a  certain  angle  to 
the  vertical,  and  that  the  vertical  plane  through  the  two  has  a 
given  inclination  to  the  vertical  north-and-south  plane.  The  given 
quantities  which  determine  the  position  are  called  the  co-ordinates 
of  the  point.  The  first  case  furnishes  an  example  of  the  ordinary 
Cartesian  system  of  rectangular  co-ordinates,  the  second  illustrates 
the  system  known  as  polar  co-ordinates.  In  the  Cartesian  system 


FIG.  20. 

the  co-ordinates  are  usually  denoted  by  the  letters  a?,  y,  z ;  in  the 
polar  system,  the  length  is  generally  denoted  by  r,  and  the  angles 


MOTION.  49 

by  9  and  0.     Thus,  in  Fig.  20,  the  relative  position  of   P  to  O 
may  be  given  by  #  =  AB,  ?/  =  OA,  2=BP;  or  by  r=OP, 


40.  Displacement.  —  When  two  points  occupy  different  positions, 
we  speak  of  the  displacement  of  the  one  from  the  other  ;  and  it  is  by 
means  of  the  displacement  that  we  determine  their  relative  position. 
Two  ideas  are  essentially  involved  in  the  term  —  the  idea  of  magnitude 
and  the  idea  of  direction.  If  we  know  only  that  one  point  is  distant 
three  feet  from  another,  we  cannot  tell  what  position  it  occupies 
on  the  spherical  surface  all  points  of  which  satisfy  this  condition, 
Other  two  conditions  are  required  to  fix  the  direction  also,  that  is,  to 
determine  the  displacement. 

Addition  of  two  displacements  is  effected  when  we  find  the  single 
displacement,  which  produces  the  same  result  as  the  two  do  when 


FIG.  21. 

applied  consecutively.  The  displacement  from  a  to  b  may  be 
denoted  by  ab,  and  that  from  6  to  c  by  be.  Then  ab-\-bc  =  ac. 
But  the  displacement  denoted  by  be  might  have  been  performed 
first.  The  effect  would  be  to  transfer  a  to  b',  which  is  a  point  such 
that  ab'  is  equal  and  parallel  to  be.  This  follows  since  a  displace- 
ment does  not  involve  the  idea  of  position,  but  only  the  ideas  of 
magnitude  and  direction ;  in  fact,  ab'  is  the  same  displacement  as 
be.  And,  similarly,  bTc  is  the  same  as  ab  ;  so  that  ab' +  brc=Tc  +  ab 
=  ac  =  ab  +  be.  Also,  since  a  displacement  is  reversed  when  its 
direction  is  reversed,  we  have  ab—-ba ;  and  the  ordinary  laws  of 
algebra  apply  to  addition  and  subtraction  of  displacements.  The 
lines  ab,  be,  etc.,  may  be  used  to  represent  the  displacements  ab, 
be,  etc. ;  for  a  line  involves  necessarily,  and  only,  magnitude  and 
direction. 

A  line,  given  in  magnitude  and  direction,  is  called  a  vector ;  and 
any  quantity  which,  like  a  displacement,  requires  for  its  com- 
plete representation  a  directed  line  is  called  a  vector  quantity.  In 
the  course  of  this  chapter,  and  of  Cha£.  Vl.»  We  shall  get  various 
examples  of  such  quantities. 

4 


50  A   MANUAL    OF   PHYSICS. 

A  quantity  which  is  independent  of  direction  —  which  merely 
possesses  magnitude — is  called  a  scalar  quantity,  that  is,  it  is  com- 
pletely determined  by  measurement  on  a  suitable  scale.  When 
treating  such  quantities  algebraically  it  is  usual  to  denote  scaiars  by 
ordinary  letters,  as  a,  6,  x,  y,  etc.,  and  vectors  by  Greek  letters  as 
a,  (3,  y,  etc. 

Any  displacement  in  space  can  be  fully  represented  in  terms  of 
three  distinct  unit  vectors  and  three  independent  scaiars.  Thus, 
in  the  figure  of  last  section,  a  may  represent  a  unit  length 
drawn  in  the  direction  AB,  and  the  line  AB  may  contain  x  units 
of  length,  so  that  the  vector  AB  is  xa.  Similarly,  if  0  is  a  unit 
vector  in  the  direction  Oy,  the  vector  OA  may  be  denoted  by  yfl  ; 
and  the  vector  BP  may  be  represented  as  zy.  Hence  -  the  vector 
OP  is  xa+y/3+zy.  But  this  quantity  denotes  merely  the  position 
of  P  relatively  to  that  of  0  :  and,  consequently,  if  any  other  point 
P'  is  situated  relatively  to  another  point  0'  in  the  same  way  in  which 
Pis  situated  with^ respect  to  0,  the  vector  xa+y$  +  zy  represents  the 
displacement  O'F. 

If  all  the  quantities  x,  y,  z,  are  variable  arbitrarily,  the  vector 

p  =  Xa  -f  yfi  +  zy 

is  the  vector  of  any  point  in  space.  If  x  and  y  alone  are  variable 
arbitrarily,  the  point  lies  upon  a  plane  parallel  to  the  directions 
indicated  by  cr,  (3.  If  one  of  the  quantities  alone  varies,  say  x,  the 
point  lies  on  a  line  whose  direction  is  indicated  by  a.  If  all  three  are 
fixed,  the  point  is  fully  determined.  More  generally,  if  x,  y,  and  « 
are  connected  by  one,  two,  or  three  relations  respectively,  the 
equation  indicates  respectively  a  surface,  a  curve,  or  a  definite  point. 

We  have  spoken  of  the  displacement  of  one  point  relatively  to 
another  as  that  which  determines  the  relative  position  in  space  of  the 
first  with  respect  to  the  second.  Sometimes  the  two  points  may  lie  on 
a  given  surface  or  a  given  curve,  and  it  is  then  frequently  convenient 
to  speak  of  the  displacement  on  the  surface  or  along  the  curve.  This 
means  that  the  magnitude  of  the  displacement  is  to  be  measured 
along  the  surface  or  curve. 

41.  Speed  and  Velocity. — The  displacement  of  a  point  may  be 
constant,  or  it  may  vary.  If  it  varies,  we  say  that  the  point  is  in 
motion :  so  that  by  motion  we  mean  change  of  relative  position.  [The 
science  which  treats  of  motion,  and  which  is  generally  called  Kine- 
matics, therefore  deals  with  the  ideas  both  of  space  and  of  time.] 
The  motion  of  a  point  is  essentially  a  translation,  for  it  has  no 
separate  parts  which  can  rotate  relatively  to  each  other.  Its  posi- 
tion, we  have  seen,  is  not  fixed  unless  three  independent  conditions 


MOTION.  51 

are  given.  The  removal  of  one  restraining  condition  leaves  the  point 
more  free  to  move  than  before  ;  and  a  point,  the  motion  of  which 
is  unrestrained,  is  said  to  have  three  degrees  of  translational 
freedom. 


FIG.  22. 

If  P  moves  to  P'  (Fig.  22),  the  change  of  displacement  is  represented 
by  thejine  PP\  or  (§  40),  by  p'— p,  where  p'  and  p  represent  the  vec- 
tors OP'  and  OP.  If  this  change  occurs  in  time  V  -  t,  the  time-rate  of 

change  is  |  /^—^  or  -jfe  It  is  convenient  to  represent  this  quan- 
tity by  p  ;  so  that  p  is  the  time-rate  of  variation  of  p,  or  its  first 
derived  function  with  respect  to  the  time  as  the  independent 
variable  ($28). 


FIG.  23. 

The  mere  magnitude  of  this  time-rate  is  called  the  Speed  of  the 
moving  point ;  but,  when  the  direction  is  considered  also,  the  term 
Velocity  is  used. 

In  illustration  of  this  we  shall  consider  the  case  of  uniform 
motion  in  a  straight  line.  Let  QP  (Fig.  23)  be  the  line.  Let  Q  be 
a  fixed  point  on  it,  and  let  P  be  the  uniformly  moving  point. 
Since  P  moves  uniformly,  the  length  of  QP  is  proportional  to  the 

4—2 


52  A    MANUAL    OF    PHYSICS. 

time  t  reckoned  from  the  instant  at  which  P  occupied  the  position 
Q  ;  say  QP  —  at,  where  a  is  a  constant.  Let  (3  be  a  unit  vector  in 
the  direction  QP,  and  let  «,  p  be  the  vectors  from  O  to  Q  and  P 
respectively.  We  have  then 

P  =  «  +  at/3. 
Hence,  by  the  principles  of  Chapter  IV., 


Here  a  is  the  speed  of  motion,  and  p  is  the  velocity.  The  magni- 
tude a  is  constant,  and  the  direction  /3  is  also  fixed.  Also  p—  a  =-.  at(3. 
Hence  the  distance  traversed  is 

s  =  at. 

[This  necessitates  our  defining  unit  speed  as  that  speed  with  which 
unit  distance  is  described  in  unit  time.] 

42.  Acceleration  of  Speed.  When  the  velocity  of  a  moving  point 
varies  it  is  said  to  be  accelerated,  and  the  time-rate  of  its  change 
is  called  the  acceleration.  Meantime  we  shall  suppose  that  the 
change  affects  the  magnitude  only  and  not  the  direction  :  that  is,  we 
shall  investigate  non-uniform  motion  in  a  straight  line.  We  may 
write  (using  the  diagram  of  last  section), 

P  =  a  +  #/3, 
where  x  is  some  function  of  t,  and  we  get 

p  =  »/3. 
Here   x  is  the  variable  speed  of  motion.     Forming  the   second 

derived  function  of  p  with  respect  to  the  time  and  denoting  it  by  p, 
we  obtain 

p  =  */3. 

If  we  assume  the  acceleration  to  be  constant,  that  is,  x  =  a  con- 
stant =  b  (say),  this  becomes 

p  =  b(3. 
But,  §  35,  this  equation  gives 

p  =  (a  +  bt)  ft+y=xft+y 

where  a  is  the  value  of  the  speed  when  t  =-  o  (usually  called  the 
initial  speed),  and  y  is  a  constant  vector  which  vanishes  if  we  sup- 
pose the  velocity  to  have  the  direction  /3.  And  from  this  we  deduce 
further 

p  =  ft/xdt  =  ft/  (a+bt)  dt  =  (c  +  at  +  |6*2)  ft  +  a  =  xfl+a. 
If  we  suppose  the  point  0  to  be  in  the  line  of  motion,  a  will  vanish, 
x  will  be  the  distance  traversed  from  Q,  and  c  will  be  the  distance 


MOTION.  53 

from  O  to  .Q.  In  other  words,  c  is  the  numerical  value  of  p  (called 
the  tensor  of  p,  and  denoted  by  the  symbol  Tp)  when  t  =  o.  Thus. 
under  uniform  acceleration  in  the  direction  of  motion  we  get 

x  =  b 

x  —  a  -\-  bt 


If  the  acceleration  is  negative,  'i.e.,  opposite  to  the  direction  of 
motion,  we  must  prefix  the  minus  sign  to  the  quantity  b.  If  the 
motion  begins  from  rest,  the  quantities  a  and  c  vanish. 

As  a  special  case,  when  a  body  falls  from  rest  under  the  action  of 
gravity,  in  which  case  the  acceleration  is  denoted  by  the  letter  g, 
we  get 

x=g,  x=gt,  x  =  \g&. 

Again,  if  the  body  is  thrown,  upwards  with  speed  V  and  we 
consider  the  upward  direction  to  be  positive,  the  equations  become 

*=-<7,  x  =  V-gt,  x^t-\gt\ 

The  second  of  these  equations  enables  us  to  tell  at  once  how  long 
the  body  will  take  to  rise  to  its  greatest  height  above  the  ground. 
For,  when  the  body  is  at  its  greatest  height,  the  speed  vanishes,  that 
is,  x  =  o,  and  therefore 

«-Y. 

9 

From  the  third  equation  we  can  tell  after  what  time  it  will  reach 
the  ground.  Since  we  are  supposing  x  to  be  measured  from  the 
surface  of  the  earth,  the  condition  is  x  =  o.  This  condition  gives 
either  t  =  o,  or 

*  =  2V. 

g 

Hence  the  body  takes  as  long  a  time  to  fall  from  its  greatest  height 
as  it  took  to  rise  to  it.  Again,  we  have  x2  =  V2-2V#£  +  #2£2  = 
V2  -  2g(Vt  -  ±g&}  =  V2  -  2gx.  When  the  point  is  at  its  greatest 
height,  x  =  o,  and 

V2 

x=  —  • 
2<7 

Also,  when  cc  =  o,  we  get  «=±V;  that  is,  the  point  reaches  the 
ground  with  speed  —V  equal  and  opposite  to  the  speed  of  pro- 
jection. 

43.  Curvature.  Acceleration  of  Velocity.  —  In  the  preceding 
section  we  have  supposed  the  direction  of  motion  to  be  unaltered. 
When  the  direction  changes  the  path  of  the  moving  point  is  said  to 


54  A   MANUAL    OF    PHYSICS. 

be  '  curved.'  The  tangent  to  the  curve  gives  the  direction  of  motion 
at  any  instant,  and  the  limiting  value  of  the  ratio  of  the  angle  between 
two  tangents  at  near  points  to  the  length  of  path  between  these 
points  as  they  are  taken  closer  and  closer  together  and  finally  coin- 
cide is  called  the  Curvature  of  the  path  at  that  place.  Thus  the 
curvature  is  the  rate  of  change  of  the  direction  of  motion  per  unit 
length  of  the  curve.  If  the  tangent  to  the  curve  at  any  point  makes 
an  angle  9  with  any  fixed  line  in  its  plane,  while  s  is  the  length  of 
the  curve,  this  definition  gives  as  the  measure  of  the  curvature  the 
quantity  dO/ds. 

To  obtain  a  measure  of  the  angle  between  two  lines  in  a  plane 
(and  we  are  here  limiting  ourselves  to  the  case  of  plane  curves) 
draw  a  circle  (Fig.  24),  of  any  radius  r,  from  the  point  of  intersection 
of  the  lines  as  centre.  The  angle  9  is  measured  by  the  ratio  which 
the  length,  s,  of  the  arc  of  the  circle  intercepted  between  the  lines 
bears  to  the  radius.  It  follows  that  the  ratio  of  9  to  s  is  equal  to  1/r, 


FIG.  24. 

and  is  therefore  constant  for  a  given  circle  no  matter  how  large  or 
how  small  9  and  s  may  be.  Hence  the  curvature  of  a  circle  is  the 
reciprocal  of  the  radius. 

Now  it  is  always  possible  to  draw  a  circle  the  curvature  of  which 
is  the  same  as  that  of  a  given  curve  at  a  given  point.  This  circle  is 
called  the  circle  of  curvature  at  that  point ;  its  radius  is  called  the 
radius  of  curvature ;  and  the  reciprocal  of  its  radius  measures  the 
curvature  of  the  given  curve  at  that  point. 

In  considering  change  of  velocity  as  dependent  on  change  of  direc- 
tion, it  will  be  convenient  to  assume  first  that  the  speed  is  constant, 
and  also  that  the  rate  of  change  of  direction  is  constant.  In  other 
words,  we  shall  investigate  the  case  of  uniform  motion  in  a  circle. 
Draw  any  two  diameters  at  right  angles  to  each  other  (Fig.  25),  and  let 
o,  (3  be  unit  vectors  along  them.  Let  p  be  the  vector  of  the  point  P. 
According  to  the  data  OP  revolves  uniformly,  that  is,  the  angle 
through  which  it  turns  per  unit  of  time  (called  the  Angular  Velocity) 
is  constant.  Let  w  be  the  angular  velocity,  so  that,  if  P  starts  from 
the  point  A,  the  value  of  the  angle  POA  is  wt,  where  t  is  the  time 


MOTION. 


55 


taken  by  the  point  to  travel  over  the  distance  AP.  Then  we  have 
vector  ON  =  OP  cos  iot  .  «,  and  vector  NP  =  OP  sin  ut  .  £,  so  that 
ifOP«=a 

p  =  a  (cos  (at  .  a  -j-  sin  at  .  (3) 

=  a  —  w  sin  cot  .  a  +  w  cos  at  .  3 


=  —  u?a  (cos  w 


sn 


This  result  shows  that  the  direction  of  the  acceleration  is  inwards 
(for  the  negative  sign  is  used)  along  OP,  and  that  its  magnitude  is 
the  square  of  the  angular  velocity  multiplied  by  the  tensor  (§  42)  of 
p.  But  the  tensor  of  p  is  a,  and  the  angle  described  per  unit  of  time  is 
the  speed,  v,  of  P  in  the  curve  divided  by  a.  Hence  w2  is  equal  to 


FIG.  25. 

v2/a2,  and  therefore  the  magnitude  of  the  acceleration  is  v2/a.  This 
acceleration,  it  is  to  be  observed,  does  not  alter  the  speed,  but  only 
the  direction  of  motion  ;  the  reason  being  that  it  is  perpendicular  to 
that  direction. 

If  OP  be   constant  in   direction,  but   of  varying  length  r,  the 

acceleration  is  r  ;  and  we  have  just  found  that,  if  r  is  fixed  in 
length  but  revolves  with  angular  velocity  w  (  =  9,  if  OP  makes  an 
angle  9  with  a  fixed  line),  the  acceleration  is-r02.  Hence,  if  both 
magnitude  and  direction  vary,  the  acceleration  along  r  is 


44.  Acceleration  in,  and  perpendicular  to,  the  Direction  of 
Motion.  —  When  a  point  moves  in  any  curve,  the  acceleration 
perpendicular  to  the  direction  of  motion  at  any  position  may  be 
found  by  drawing  the  circle  of  curvature  at  the  given  position.  If 
r  is  the  radius  of  curvature,  and  v  the  speed  of  motion,  the  accelera- 
tion perpendicular  to  the  direction  of  motion  is,  by  last  section,  vzfr 

towards  the  centre  of  curvature,  and  the  acceleration  of  speed  is  s 


56  A   MANUAL    OF   PHYSICS. 

where  s  is  the  distance  measured  along  the  curve,  in  the  direction 
of  motion,  from  a  fixed  point  to  the  moving  point. 

The  following  method  of  deducing  these  results  is  exceedingly 
simple  and  instructive.  Let  p  be  the  vector  to  any  point  of  the 
path.  Then 

dp     dp     ds     dp  •  .   ,       N 

'=3hto"3i-5  •'•"''  (say)- 

Now,  dp  being  a  vector,  p'  is  also  a  vector  in  the  same  direction — 
that  is,  along  the  tangent  (see  Fig.  22,  in  which  PP'  must  be  supposed 
to  be  indefinitely  small).  But,  in  the  limit,  when  ds  vanishes,  the 
length  of  dp  is  equal  to  ds.  Therefore  p'  is  a  unit  vector  along  the 
tangent.  And,  since  the  length  of  p'  is  constant,  dp'  must  be  per- 
pendicular to  p'.  Hence,  dp'lds  =  p"  is  a  vector  inwards  along  the 
normal  to  the  curve.  And  the  quotient  of  the  length  of  dp'  by  the 
length  of  p'  is  equal  to  the  angle,  dO,  turned  through  by  the  radius 
of  curvature.  Hence  the  magnitude  of  p"  is  d9jds  ;  that  is,  (§  43,) 
it  is  the  reciprocal  of  the  radius  of  curvature.  But 

p  =  <yp'_j_-yp'  =  Vp'-\-V*p". 

Hence,  the  total  acceleration  is  made  up  of  a  part  v  along  the 
tangent,  and  a  part,  proportional  conjointly  to  v2  and 'to  the 
reciprocal  of  the  radius  of  curvature,  inwards  along  the  normal. 

45,  Average  Speed  and  Velocity. — If  a  point  passes  over  a  certain 
distance  in  a  certain  time  with  varying  speed,  it  is  always  possible 
to  find  a  uniform  speed  with  which  the  same  distance  might  have 
been  described  in  the  same  time.     This  is  called  the  Average  Speed 
of  the  point.     The  last  equation  of  §  41  obviously  applies  to  the 
case  of  average  speed. 

When  the  acceleration  of  speed  is  uniform,  the  average  speed  is 
clearly  half  the  sum  of  the  initial  and  final  speeds  during  the  time 
considered.  We  may  test  this  by  the  equations  of  section  42. 
V  is  the  initial  speed  of  projection,  and  V  -  gt  is  the  final  speed. 
Hence  the  average  speed  is  V—  ^gt.  Therefore,  by  the  last  equation 
of  §  41,  the  distance  x  is  equal  to  (V—$gt)t,  which  agrees  with  the 
result  already  obtained. 

The  same  results  evidently  hold  for  the  corresponding  angular 
quantities  ;  for,  in  §  42,  x  might  be  an  angular  distance. 

46.  Resolution  and  Composition  of  Velocities  and  Accelerations. 
— Velocities  and  accelerations,  since  they  are  vector  quantities  (§  40), 
are  to  be  compounded  and  resolved  in  accordance  with  the  laws  of 
composition  and  resolution  of  these  quantities.     Hence,  if  a  point  is 
subjected  to  a  series  of  simultaneous  velocities  which  are  represented 
by  all  the  sides  AB,  BC,  etc.  (Fig.  26),  of  a  closed  polygon,  taken  in  the 


MOTION.  57 

same  order  round,  except  one,  the  resultant  velocity  is  represented  by 
the  remaining  side  taken  in  the  opposite  direction  round.  This  theorem 
is  known  as  the  '  polygon  of  velocities.'  It  follows  that,  if  the  various 
velocities  to  which  a  point  is  subjected  are  representable  by  all  the 
sides  of  a  closed  polygon  taken  in  order,  the  point  is  at  rest.  For  the 
resultant  of  all  but  one  is  equal  and  opposite  to  that  one. 

Inthe  particular  case  of  two  velocities  AB  and  EC,  the  resultant 
is  AC — the  third  side  taken  in  the  opposite  direction  round.     This 


FIG.  26.  FIG.  27. 

theorem  is  known  as  the  '  triangle  oj_yelocities.'  But,  since  the 
vector  AD  is  identical  with  the  vector  BC  (Fig.  27),  we  may  say  that 
the  resultant  of  two  velocities  represented  by  adjacent  sides  of  a 
parallelogram  is  the  diagonal  drawn  from  the  same  point.  In  this 
form  the  theorem  is  known  as  the  '  parallelogram  of  velocities.' 

In  order  to  resolve  a  velocity  into  any  number  of  components 
we  have  merely  to  reverse  the  above  process.     The  problem  is 


FIG.  28. 

determinate  if  we  are  given  2(n  — 1)  conditions,  where  n  is  the 
number  of  components  and  n  of  the  conditions  are  directional. 

As  a  particular  case,  if  we  have  to  find  the  resolved  part  of  a 
velocity  AC  (Fig.  28)  in  a  direction  AB  making  an  angle  9  with  AC, 
we  require  to  know  first  the  direction  of  the  other  component.  It 
is  usually  understood  that  the  other  component  is  to  be  at  right 
angles  to  the  first,  in  which  case  we  have  AB  =  AC  cos  9. 


58 


A    MANUAL    OF    PHYSICS. 


These  remarks  apply  to  accelerations  and  to  any  other  vector 
quantities. 

47.  Motion  of  Projectiles.  —  Let  a,  /3,  be  unit  vectors  in  the  hori- 
zontal and  vertical  directions  respectively,  and  let  a  point  be  pro- 
jected from  0  (Fig.  29)  with  velocity  aa  +  bfi.  If  P  be  the  position 
of  the  point  at  time  £,  p  and  p'  being  the  components  of  the  vector 
from  0  to  P,  we  have 

p  =  ata 


In  the  latter  equation,  the  first  term  represents  the  vector  height 
to  which  the  point  would  have  ascended  had  gravity  not  acted  ;  and 


ot. 


FIG.  29. 


the  second  term  represents  the  extent  (§  42)  to  which  gravity  has 
diminished  this  height.  The  length  of  p'  is  therefore  bt-±gt*,  and 

this  vanishes  when  t  =  o  and  £=  —  •     The  value  t  =  o  corresponds  to 

the  instant  of  projection,  and  the  other  value  gives  the  time  of  flight 
on  a  horizontal  range.  Again 

p'=  6/3-^/3. 

This  vanishes  when  t  =  b/g.  But  p'  ceases  to  alter  in  magnitude  just 
at  the  instant  that  the  highest  point  of  the  path  is  reached.  Hence, 
big  is  the  time  taken  to  reach  the  greatest  height,  and  is  equal  to 
one-half  of  the  time  of  flight. 

If  this  value  of  t  be  put  in  the  expression  for  p',  we  find  that  the 
total  height  reached  is  b*/2g.  Also,  by  putting  t  =  2b/g  in  the  ex- 
pression for  p,  we  find  that  the  total  range  along  a  horizontal  line 
is  Qab/g. 

48.  The  Hodograph. — If,  from  any  point  as  origin,  a  line  be 
drawn  to  represent  the  velocity  of  a  moving  point,  the  free  extremity 
of  the  line  traces  out  a  curve  which  is  called  the  hodograph  of  the 


MOTION. 


59 


path  of  the  moving  point.  The  tangent  to  the  hodograph  gives  the 
direction  of  acceleration  in  the  path. 

When  the  hodograph  and  the  law  of  its  description  are  given,  the 
path  and  the  law  of  its  description  can  be  found.  Thus,  in  the  path 
of  a  projectile,  we  have  as  above  OP  =  (T=p+p'  =  a^a+(&£-  %gt-)(B ; 
and  therefore  (§  41)  the  vector  in  the  hodograph  is  a  =  aa+(b  -gt)(3. 
It  is  consequently  a  vertical  straight  line  uniformly  described.  And 
from  this  latter  equation  the  former  (which  is  its  integral)  can  be 
obtained. 

49.  Moments. — The  moment  of  any  quantity  is  the  measure  of 
its  importance  with  regard  to  the  production  of  some  effect.  The 


FIG.  30. 

moment  of  any  directed  quantity  (which  may  be  indicated  by  the 
line  AB)  with  reference  to  revolution  about  a  point  0  is  proportional 
to  its  own  magnitude  and  to  the  distance  of  0  from  its  line  of  action. 
If  we  define  unit  moment  as  the  moment  of  a  directed  quantity  of 
unit  magnitude  about  a  point  at  unit  distance  from  its  line  of  action, 
X>a  is  the  magnitude  of  the  moment  of  a  quantity  containing  a  units 
about  a  point  distant  p  units  of  length  from  its  line  of  action.  Thus 
the  moment  of  AB  about  0  is  twice  the  area  of  the  triangle  AOB. 

The  moment  of  the  resultant  of  two  directed  quantities  is  the 
sum  of  the  moments  of  the  components. — Let  AC,  AB  (Fig.  31),  be 
the  two  components ;  AD  being  the  resultant.  We  have  to  prove 
that  AOD  =  AOB+AOC.  Draw  OF  parallel  to  CA  to  meet  BA  and 
DC  produced  in  F  and  E.  Then  AOD  =  AOB+BOD  -  ABD  = 
AOB+i  FEDB-I  ACDB=AOB+i  FECA=AOB+AOC. 

The  lines  AC  and  AB  in  the  figure  have  been  so  drawn  that 
motion  along  them  from  A  involves  revolution  in  the  same  direction 
about  0.  Had  AC  been  so  drawn  as  to  involve  rotation  about  0 
opposite  to  that  indicated  by  AB,  it  would  have  been  necessary  to 
regard  one  of  the  triangles  as  being  negative.  The  same  proof 
would  then  hold.  It  is  usual  to  regard  rotation  opposite  to  that  of 
the  hands  of  a  watch  as  positive. 


60  A    MANUAL    OF    PHYSICS. 

When  the  direction  of  the  quantity  passes  through  0,  its  moment 
about  0  vanishes. 

50.  Acceleration  Perpendicular  to  Radius-vector. — In,  §  43  we 
obtained  an  expression  for  the  accleration  along  the  radius-vector 
of  a  moving-point.  We  can  now  find  an  expression  for  the  accelera- 
tion perpendicular  to  the  radius-vector. 

If  AB,  in  the  last  figure,  represents  the  path  of  a  moving  point  P, 
the  moment  of  the  velocity  of  P  is  twice  the  rate  at  which  the  area 
AOP  is  described  as  P  moves  along  AB.  For  if  8s  is  a  small  length 

measured  along  the  path,  pSs=p— St  is  twice  the  corresponding  in- 

ot 

crease  of  the  area  (  =  da,  say).     Therefore  77  —PT1>  and,  in  the  limit 

when  St  =  o,  ~r,=p-T.=pv,  where  v  is  the  speed,  which  proves  the 

statement.    It  is  evident  that  the  proof  still  holds  when  the  direction 
of  motion  varies,  for  it  is  true  however  small  Ss  and  £t  may  be. 

Let  OP(  =  r),  the  radius-vector  of  the  moving  point  P  (Fig,  31), 
make  an  angle  9  with  a  fixed  line  in  the  plane  of  the  figure.  Let 


FIG.  31. 

P  move  to  a  point  P',  so  near  to  P  that  PP'  is  practically  a  straight 
line.  Draw  PM  perpendicular  to  OP'.  Then  PM  =  rd0,  and 
0PM  =  £r .  rdO.  Hence  the  area  traced  out  per  unit  of  time  is 
%r-0,  so  that  r2^  expresses  the  moment  of  the  velocity.  Now  we  have 
just  seen  that  the  moment  of  the  velocity  is  equal  to  pv.  But 
pv  is  equal  to  rw,  where  u  is  the  resolved  part  of  the  velocity 
perpendicular  to  r.  This  is  evident  since,  if  p  and  r  are  inclined  at 
an  angle  $  to  each  other,  we  have  p  =  r  cos  0,  and  u  =  v  cos  0.  Hence 
we  have 

ru  =  r*V. 

Suppose  now  that  the  velocity  is  accelerated.  The  moment  of  the 
velocity  represents  the  rate  of  increase  of  the  area,  and  so  the 
moment  of  the  acceleration  gives  the  rate  of  change  of  da/dt. 


MOTION. 


61 


The  acceleration  can  be  regarded  as  composed  of  two  parts  —  one 
part  perpendicular  to  r,  and  the  other  along  r.  Of  these  components 
the  former  alone  is  effective  in  altering  the  rate  of  description  of 
the  area,  for,  §  49,  the  direction  of  the  latter  part  passes  through  0. 
Hence  the  moment  of  the  acceleration  is  rut  and  so  the  acceleration 
perpendicular  to  the  radius-vector  is 


51.  Simple  Harmonic  Motion.  —  When  a  point  P  revolves  with 
uniform  speed  in  a  circle,  the  motion  of  the  foot  N  (Fig.  32)  of  the 
perpendicular  drawn  from  P  to  any  fixed  diameter  is  called  simple 
harmonic  motion.  The  velocity  and  acceleration  of  the  point  N  can 
easily  be  found  when  the  position  and  velocity  of  P  is  given. 


FIG.  32. 


The  velocity  of  N  is  evidently  the  resolved  part  of  the  velocity  of  P 
in  the  direction  ON.  That  is  to  say,  its  magnitude  is  v  cos  0  if  v  is 
the  speed  of  P.  But  cos  9  is  proportional  to  NP ;  hence  the  speed 
of  N  is  proportional  to  NP.  The  maximum  speed  is  attained  when 
N  passes  through  O ;  and  it  is  then  equal  to  v,  the  speed  in  the 
circle. 

Similarly  the  acceleration  of  N  is  the  resolved  part  of  the 
acceleration  of  P  in  the  line  ON.  But  the  acceleration  of  P  is, 
§  43,  -  -y2/a,  where  a  is  the  radius.  Hence  the  acceleration  of  N 
is  -  v*la .  sin  9  =  —  -y2  ON/a/2.  That  is  to  say,  it  is  inwards  towards  0, 
the  centre  of  the  range  of  N  ;  and  its  magnitude  is  proportional  to 
ON,  the  displacement  from  the  centre. 

The  ratio  of  the  acceleration  to  the  displacement  is  v2/a>2  =  w2, 
where  w  is  the  angular  velocity  of  OP.  But  the  angular  velocity 


62  A   MANUAL    OF    PHYSICS. 

is  27T/7-,  T  being  the  time  of  a  complete  revolution  in  the  circle. 
Hence 

acceleration      4?r2 

displacement  ~~  ~r*  ' 

If  we  call  B  the  positive  extremity  of  the  range,  the  angle 
through  which  OP  has  turned  since  it  coincided  with  OB  is  called 
the  Phase  of  the  simple  harmonic  motion.  The  phase  may  also  be 
measured  in  fractions  of  a  whole  revolution.  Thus  we  speak  of  the 
quarter  phase,  etc. 

The  maximum  distance  to  which  N  can  get  from  0  is  called  the 
Amplitude  of  the  motion.  It  is  obviously  the  radius  of  the  cor- 
responding circle. 

If  the  motion  does  not  commence  at  the  positive  extremity  of 
the  range,  the  angle  through  which  the  radius  has  to  turn  until 
P  reaches  the  positive  end  is  called  the  Epoch.  Thus  the  general 
expression  for  x,  the  distance  of  N  from  0,  is 

x  =  a  cos  (w£+a). 

Here  a  is  the  amplitude,  w  is  the  (constant)  angular  velocity,  t  is 
the  time,  and  a  is  the  epoch. 

Simple  harmonic  motion  is  frequently  exemplified  in  nature.  It 
occurs  in  the  vibration  of  stretched  strings,  the  agitation  of  the 
luminiferous  medium,  etc. 

52.  Composition  of  Simple  Harmonic  Motions. — (1)  Motions  in 
the  same  straight  line  and  of  equal  periods.  Let  the  motion  of  P 
(Fig.  33)  correspond  to  one  of  the  given  motions.  From  P  draw 
a  line  PQ,  making  an  angle  0,  equal  to  the  phase  of  another  of  the 
motions,  with  the  line  OA ;  and  let  the  length  of  PQ  be  equal  to 
the  amplitude  of  this  motion.  Since  0  and  9  increase  at  the  same 
rate,  the  line  OQ  remains  of  constant  length  and  revolves  at  the  same 
rate  as  OP  andPQ.  But  the  foot  of  the  perpendicular  from  Q  on 
OB  moves  with  a  motion  which  is  compounded  of  the  two  given 
motions.  Hence  the  resultant  of  the  two  motions  is  another  simple 
harmonic  motion,  of  the  same  period,  in  the  same  straight  line. 
And,  in  particular,  when  the  amplitudes  of  the  two  components  are 
equal,  the  phase  of  the  resultant  is  the  mean  of  the  phases  of  the 
components. 

This  proof  is  quite  general,  and  applies  to  any  number  of  such 
simple  harmonic  motions. 

Another  proof  may  be  obtained  as  follows.  Let  the  separate 
motions  be  xl  —  a1cos(w^  +  aj,  xc)  =  a.,cos(wt  -f-  a2),  etc.  Then 
a?  =  a?14-JJa+  etc.  =aLco$(ut  +  ctj)  -f-  a.,cosU£  +  a.J+  etc.  But 


MOTION.  63 

cos  (u)t  -f  a1)  =  cos  u>t  cos  ttj  —sin  wt  sin  av  Therefore  x  =  (al  cos  ax  +  a2 
cos  a.2-^-  etc.)  cos  w£  —  (%  sin  aj-|-#2  sin  a24-  etc.)  sin  wt.  Now  assume 
the  multiplier  of  cos  ut  to  be  equal  to  a  cos  a,  and  the  multiplier  of 
sin  at  to  be  a  sin  «,  so  that  x  =  a  cos  w£  cos  a  -  a  sin  wf  sin  a.  This 
gives  x  =  a  cos  (>ot-\-a).  [The  assumption  made  is  obviously  justifi- 
able ;  for  it  only  introduces  two  new  quantities  a  and  a,  and  gives  two 
equations  to  determine  their  values.]  Hence,  in  this  case,  the 
resultant  is  simple  harmonic  motion  of  the  same  period  in  the  same 
straight  line. 

(2)  Two  simple  harmonic  motions,  of  the  same  period  and  phase, 
in  lines  inclined  at  any  angle.  These  obviously  compound  into 
a  single  motion  of  the  same  period  and  phase.  For,  let  OA,  OB 
be  the  two  inclined  lines,  and  draw  any  other  line  OP.  From  P  draw 
PM,  PN,  parallel  to  OA,  OB,  respectively,  to  meet  these  lines  in  the 
points  N  and  M  respectively.  If  P  moves  along  OP  with  simple 


0  N  A 

FIG.  33. 

harmonic  motion,  it  is  evident  that  M  and  N  move  along  OB  and 
OA  with  simple  harmonic  motions  of  the  same  period  and  phase. 
And  the  motion  of  P  is  the  resultant  of  their  motions. 

(3)  Two  simple  harmonic  motions  in  lines  at  right  angles  to  each 
other  of  the  same  period  and  amplitude,  and  differing  in  phase  by 
one  quarter  of  a  period.    A  glance  at  the  figure  of  last  section  shows 
that  M'  moves  along  OA'  precisely  as  N  moves  along  OB,  since  OP' 
is  perpendicular  to  OP.     Hence  the  motions  of  N  and  M  differ  in 
phase  by  one  quarter  of  a  period.     And  the  motion  of  P  is  their 
resultant.     That  is  to  say,  the  resultant  is  uniform  circular  motion  ; 
and  the  motion  takes  place  from  the  positive  end  of  the  range  in 
which  the  motion  is  one  quarter   of  a  period  in  advance  to  the 
positive  end  of  the  other  range. 

(4)  Two  simple  harmonic  motions  of  equal  period,  and  of  phases 
differing  by  a  quarter  period,  in  lines  inclined  at  any  angle.     By 
projection  of  the  circle  we  obtain  an  ellipse  in  which  conjugate 
diameters  correspond  to  mutually  perpendicular  diameters  of  the 
circle.     Hence   the  resultant  is  elliptic  motion,  with  equal  areas 


64  A   MANUAL   OF   PHYSICS. 

described  in  equal  times  (since  this  is  so  in  the  circle),  and  with  a 
law  similar  to  that  given  in  (3)  as  regards  direction  of  motion. 

(5)  Any  number  of  simple  harmonic  motions,  in  lines  inclined  at 
any  angle  to  each  other,  and  of  any  phase,  but  of  equal  periods. 
By  a  reversal  of  the  first  proof  of  (1),  we  see  that  the  line  OQ 
may  be  replaced  by  any  two  lines  OP  and  PQ,  which  revolve 
with  the  same  angular  velocity.  Hence  any  simple  harmonic 
motion  may  be  broken  up  into  two  of  the  same  period,  which 
differ  in  phase  by  any  given  amount,  and  one  of  which  has  any 
given  phase.  This  appears  also  by  a  reversal  of  the  second  proof 
of  (1).  For  if  a  cos  (<at  +  a)  is  to  be  identical  with  a:  cos  (<ot  +  «i) 
+  a.2  cos  (wt  -f  a2),  we  must  have  ax  cos  ^  +  a.2  cos  cr2  =  a  cos  «>  an(i 
^sinaj  -}-  a2  sina2  =  a  sin  a.  That  is,  there  are  only  two  conditions 
to  be  satisfied  by  the  four  quantities  av  av  alt  a2 ;  so  that  two 
more  may  be  imposed. 

Let  Pj,  P2,  etc.  (Fig.  34),  be  the  points  moving  with  simple 
harmonic  motion.  Let  pi,  p\,  be  two  points,  whose  motions  com- 


FIG.  34. 

pound  into  that  of  PI  ;  and  let  their  phases  differ  by  a  quarter  period. 
D.eal  similarly  with  P2,  etc.,  and  let  the  motions  of  p^  p^  etc.,  be 
all  of  the  same  phase,  while  those  of  p\,  p'2,  etc.,  also  agree  in 
phase  with  each  other,  but,  of  course,  differ  in  phase  from  the 
motions  of  pi,  p^,  etc.,  by  a  quarter  of  a  period.  Besolve  all 
these  motions  into  their  components  along  two  rectangular  axes 
ox  and  oy.  Then  compound  all  of  the  same  phase  in  each  axis 
with  each  other.  The  result  is,  by  (1),  two  simple  harmonic 
motions  in  each  axis  differing  in  phase  by  a  quarter  period.  Now 
combine  each  motion  in  oy  with  the  motion  in  ox  which  is  of  the 
same  phase,  and  we  have  ultimately  two  simple  harmonic  motions 
which  differ  in  phase  by  a  quarter  period  and  take  place  in  lines 
which  are  in  general  inclined  to  each  other  at  a  finite  angle,  These, 
as  we  have  seen,  combine  into  elliptic  motion. 


MOTION.  65 

53.  Wave  Motion  along  a  Line. — Let  a  point  vibrate  up  and 
down  the  ?/-axis  with  simple  harmonic  motion  about  the  origin, 
and  let  the  paper  be  drawn  along  behind  it  at  a  uniform  rate  in  the 
direction  xo.  The  point  will  trace  out  a  curve  (indicated  in  the 
figure)  which  exhibits  the  simplest  form  of  a  wave.  The  quantities 
y  and  x  can  easily  be  seen  to  be  connected  by  the  relation 

y  =  a  cos  (ut  -  nx] 

in  which  a  and  M  denote  the  same  quantities  as  formerly.  If  x  is 
constant,  the  equation  is  of  the  same  form  as  the  one  which  was 
given  in  §  51,  and  shows  that  every  point  on  the  a>axis  vibrates  up 


and  down  with  simple  harmonic  motion,  and  that  y  has  one  and 
the  same  value  for  all  the  values  of  t  which  differ  by  the  amount 
27r/w.  This  quantity  STT/W  is  called  the  periodic  time,  or  the  period, 
of  the  motion. 

If  t  remains  constant,  y  varies  with  x  in  precisely  the  same 
manner  as  it  did  when  x  was  constant  and  t  varied.  The  value  of  y 
is  the  same  for  all  values  of  x  which  differ  by  Sirln.  This  quantity 
is  the  Wave-length. 

Lastly,  y  remains  fixed  in  value  if  x  and  t  vary  simultaneously  in 
such  a  way  that  (oit  -  nx}  is  zero,  x  being  measured  from  any  special 
position,  and  t  from  any  definite  instant.  This  gives  x  =  w/ra,  and 
shows  that  the  wave  runs  along  in  the  direction  ox  with  speed  w/n 
and  without  change  of  form. 

Similar  reasoning  shows  that  the  equation 

y=a  cos  ((ut  +  nx)- 

represents  a  succession  of  precisely  similar  and  equal  waves  which 
run  in  towards  the  origin  with  speed  —  w/n.  The  resultant  disturb- 
ance due  to  the  superposition  of  this  set  of  waves  upon  the  former 
is  represented  by 

y—a  {cos  (ut -nx)  +  cos  (o)t  +  nx}\ 

=%a  cos  (tit  cos  nx. 
Whatever  be  the  value  of  x,  this  vanishes  when  t  is  any  odd  multiple 


66 


A    MANUAL    OF    PHYSIC8. 


of  ir/2w ;  and,  whatever  be  the  value  of  t,  it  vanishes  when  x  is  any 
odd  multiple  of  7r/2w.  The  motion,  at  any  definite  point  is  simple 
harmonic,  of  period  27r/w ;  and  the  form  of  the  wave  at  any  definite 
instant  resembles  that  shown  in  Fig.  35,  the  ordinates  being  all 
altered  in  the  common  ratio  of  2  cos  w£  to  unity.  The  resultant  is 
therefore  a  series  of  stationary  waves,  which  oscillate  up  and  down, 
parallel  to  the  ?/-axis,  in  situ.  This  result  is  of  importance  in  the 
theory  of  the  vibrations  of  stretched  strings,  etc. 

54.  Rotation. — While  a  mere  point  can  have  translational  motion 
only,  a  rigid  body  (a  body  the  parts  of  which  cannot  suffer  relative 
displacement)  is  free  to  rotate  also  unless  three  points  of  it,  which 
do  not  lie  in  the  same  straight  line,  are  fixed.  Three  such  points 
being  fixed,  the  body  is  devoid  of  all  freedom  to  move.  If  two 
points  are  fixed  it  can  rotate  about  the  line  which  joins  them,  and 
is  said  to  have  one  degree  of  rotational  freedom.  If  one  point  only 
is  fixed  the  body  may  rotate  independently  about  any  three  mutually 
perpendicular  axes  which  pass  through  that  point — it  has  three 
degrees  of  rotational  freedom.  Finally,  no  point  being  fixed,  it 
lias,  in  addition,  three  degrees  of  translational  freedom.  The 
greatest  number  of  degrees  of  freedom  which  a  rigid  body  can  have 
is  therefore  six.  [A  non-rigid  body  has  distortional  freedom  also.] 

,  55.  Alteration  of  Co-ordinates  because  of  Rotation. — If  a  point 
P  is  rotating  about  the  axis  of  z  (drawn  perpendicular  to  the  plane  of 
the  paper  through  the  point  O,  Fig.  36)  with  angular  velocity  w.,  the 


FIG.  36. 


FIG.  37. 


alteration  of  the  x  co-ordinate  of  P,  in  the  time  ct,  is  -  wzr  cos  ^ct  — 
—  atzr  sin  0  St  =  -  Mzy$t.  If  P  were  revolving  simultaneously  about 
Oy  with  angular  velocity  wy,  the  alteration  of  x  in  the  same  time 
would  be  MyZdt.  Hence  the  resolved  part  of  the  speed  of  P  parallel 
to  Ox  is  (dt  being  small)  ajyz  -  wz?/. 

56.  Uniplanar  Motion  of  a  Rigid  Body. — By  '  uniplanar  motion  ' 
is  meant  motion  parallel  to  one  plane.  The  motion  of  a  rigid 
plane  figure  is  included  as  a  special  case. 


MOTION.  67 

Let  the  motion  be  parallel  to  the  plane  of  the  paper;  and  let 
AB  be  the  position  of  a  line  in  the  body  before  the  motion  occurs, 
while  A'B'  is  its  position  at  the  end  of  the  motion.  Draw  AA'  and 
BB' ;  bisect  them,  and  erect  perpendiculars  at  their  points  of  .bi- 
section. Let  these  meet  in  O.  We  have  OA  =  OA',  and  OB  =  OB'. 
Also  AOB  and  A'OB'  are  congruent  triangles,  the  angle  AOB  being 
equal  to  the  angle  A'OB'.  Thus  AB  might  have  been  moved  into 
the  position  A'B'  by  a  single  rotation,  about  O  as  centre,  through 
the  angle  AOA'.  Hence  any  displacement  of  a  rigid  body  parallel 
to  one  plane  may  be  produced  by  rotation  about  a  definite  axis 
perpendicular  to  that  plane. 

In  general  the  body  does  not  revolve  in  this  way  from  its  initial 
to  its  final  position.  On  the  contrary,  each  point  usually  describes 
a  curve  which  is  not  the  arc  of  a  circle.  In  such  a  case  we  may 
regard  the  total  displacement  of  any  point  as  made  up  of  a  succession 
of  indefinitely  small  displacements,  each  of  which  coincides  with  an 
indefinitely  small  arc  of  a  circle.  This  circle  is  evidently  the 
circle  of  curvature  (§  43)  of  the  path  of  the  moving  point.  Its 
centre  is  called  the  instantaneous  centre  about  which  all  points  in 
the  same  plane  are  revolving.  Thus,  when  a  wheel  rolls  along  the 
ground,  the  point  of  the  wheel  which  is  in  contact  with  the  ground 
is  at  rest  for  an  instant — it  is  the  instantaneous  centre  about  which 
the  wheel  is  revolving  for  a  moment  as  a  rigid  body. 

Let  the  point  O,  Fig.  38,  be  the  instantaneous  centre.  The  point  pl 
revolving  about  0  will  come  into  the  position jp'*.  Suppose  now  that  p., 
revolves  about  p\  as  the  new  instantaneous  centre,  and  that  this 
brings  it  into  coincidence  with  p'.2  about  which  the  revolution  next 


takes  place,  and  so  on.  The  points  plt  p.21  etc.,  are  points,  fixed  in 
the  body,  which  are  successively  at  rest,  and  the  points p\,  p'%,  etc., 
are  points,  fixed  in  space,  at  each  of  which  in  succession  the  in- 
stantaneous centre  is  situated.  The  instantaneous  centre  coincides 
for  a  short  time  with  each  point  of  both  series,  and  passes  suddenly 

5—2 


68  A    MANUAL    OF    PHYSICS. 

from  one  to  another.  When  the  motion  is  continuous  the  polygons 
°PiP-2  •  •  •  and  op'ip'2  .  .  .  become  continuous  curves,  and  the 
instantaneous  centre  moves  continuously  along  them.  In  the  case 
of  a  rigid  body,  moving  parallel  to  the  plane  of  the  paper,  the  line 
through  the  centre  perpendicular  to  that  plane  is  instantaneously  at 
rest,  and  is  called  the  instantaneous  axis  ;  and  the  curves  in  the 
figure  are  sections  of  cylindrical  surfaces  in  the  body  and  in  space. 
Hence  we  see  that  the  most  general  uniplanar  motion  of  a  rigid 
body  consists  in  the  rolling  of  a  cylinder  fixed  in  the  body  upon  a 
cylinder  fixed  in  space.  An  obvious  example  of  this  is  given  when 
a  roller  is  drawn  over  the  surface  of  the  ground. 

A  similar  statement,  modified  merely  by  the  substitution  of  the 
word  '  curve  '  for  '  cylinder,'  applies  to  the  motion  of  a  plane  figure 
in  its  own  plane. 

Mere  translation  is  a  special  case  in  which  the  instantaneous 
centre  is  at  an  infinite  distance.  It  may  be  considered  to  consist  in 
infinitely  slow  rotation  about  an  infinitely  distant  axis. 

57.  Motion  of  a  Rigid  Body  in  Space.- — First,  suppose  one  point 
of  the  body  to  be  fixed,  and  consider  a  sphere  in  the  body  with  its 
centre  at  the  fixed  point.  Take  any  two  points  A,B,  on  the  surface 
of  this  sphere  which  occupy  the  positions  A',B',  respectively,  at  the 
end  of  the  motion.  The  reasoning  of  last  section  applies  here  also, 
great  circles  of  the  sphere  taking  the  place  of  straight  lines.  We 
thus  find  that  the  displacement  might  have  been  produced  by  simple 
rotation  of  the  sphere  about  a  diameter  passing  through  the  point  O 
(Fig.  37)  on  the  surface. 

The  actual  motion  consists  in  the  rolling  of  a  cone  fixed  in  the 
body  upon  a  cone  fixed  in  space.  This  is  at  once  evident  if  we 
suppose  the  curve  op^.^  ...  of  last  section  to  be  drawn  upon  the 
surface  of  the  sphere  whose  centre  is  fixed — the  points  pit  pz,  etc., 
being  successive  positions  of  the  extremity  of  the  instantaneously 
fixed  diameter. 

Now  suppose  that  no  point  is  fixed.  The  total  displacement 
consists  in  general  of  both  translational  displacement  and  rotational 
displacement ;  and  it  is  clear  that  we  may  separate  these,  taking 
first  one  and  then  the  other,  and  yet  produce  the  same  total  effect. 
As  we  have  just  seen,  the  rotation  leaves  a  set  of  planes  (those 
perpendicular  to  the  axis  of  rotation)  parallel  to  their  original 
position,  and  mere  translation  does  not  alter  this  parallelism  ;  so 
that,  in  the  final  position  of  the  body,  there  is  one  set  of  planes  which 
has  been  unaltered  as  regards  orientation.  The  required  rotational 
displacement  can  be  produced  by  revolution  about  any  axis  perpen- 
dicular to  these  unaltered  planes,  but  the  final  position  of  the  body 


MOTION.  69 

will  depend  on  the  particular  axis  chosen.  And  we  can  so  choose 
the  axis  that,  after  the  rotation  has  taken  place,  mere  translation 
parallel  to  that  axis  will  make  the  body  take  the  required  position. 
And  this  can  only  be  done  in  one  way.  For  let  the  plane  of  the 
paper  be  one  of  the  planes  which  are  unaltered  in  direction,  and  let 
AO  be  a  line  in  that  plane,  the  final  position  of  which  is  to  be  A'O'  in 
a  parallel  plane.  Kotation  through  the  angle  between  AO  and  A'O', 


FIG.  39. 

about  any  point  in  the  plane  of  the  paper,  will  make  AO  parallel  to 
A'O' ;  but  AO  will  only  be  superposed  upon  A'O'  if  the  point  be 
chosen  by  means  of  the  construction  of  last  section.  And,  when 
this  superposition  is  effected,  a  translation  perpendicular  to  the  plane 
of  the  paper  will  bring  the  body  into  its  final  position.  Hence 
any  displacement  of  a  rigid  body  in  space  may  be  produced  by 
rotation  about,  and  translation  along,  a  definite  axis. 

When  the  rotation  and  the  translation  are  simultaneous,  the 
motion  is  called  a  twist  about  a  screw.  Such  a  motion  is  the  most 
general  kind  of  motion  that  a  body  which  possesses  only  one  degree 
of  freedom  can  have.  [The  rotation  and  the  translation  do  not 
occur  independently,  and  so  one  degree  of  freedom  alone  is 
involved.] 

Any  given  motion  of  a  rigid  body  in  space  consists  in  a  twist 
about  a  screw,  in  which  the  axis  and  the  linear  and  angular 
velocities  are  in  general  varying.  The  position  of  the  axis  at  any 
instant  is  given  by  the  line  of  contact  of  two  ruled  surfaces,  one  of 
which  (fixed  in  the  body)  simultaneously  rolls  and  slides  upon  the 
other,  which  is  fixed  in  space. 

58.  Composition  of  Angular  Velocities. — An  angular  velocity  is 
completely  determined  when  its  magnitude  and  direction  are  given, 
and  these  quantities  may  be  indicated  by  the  magnitude  and  direc- 
tion of  a  line  drawn  parallel  to  the  axis  of  rotation.  Angular 
velocities  (and  accelerations)  are  therefore  vector  quantities,  and 
their  laws  of  composition  and  resolution  are  identical  with  the  laws 
for  vectors,  and  therefore  with  the  laws  for  linear  velocities  and 


70  A    MANUAL    OF    PHYSIOS. 

accelerations.  Thus  the  resultant  of  two  angular  velocities  or 
accelerations,  which  are  represented  by  the  two  sides  of  a  parallelo- 
gram, is  represented,  on  the  same  scale,  by  the  conterminous 
diagonal. 

An  extremely  important  case  is  that  in  which  a  body  with  one 
point  fixed  is  revolving  uniformly  about  an  axis  and  is  subjected  to 
uniform  angular  acceleration  about  a  perpendicular  axis.  In  the 
corresponding  problem  regarding  linear  velocity  and  acceleration 
(§  43),  the  magnitude  of  the  linear  velocity  is  unaltered  while  the 
direction  of  motion  revolves  uniformly,  being  always  perpendicular 
to  the  direction  of  acceleration,  which  also  revolves  uniformly.  So, 
in  the  present  case,  we  can  at  once  assert  that  the  magnitude  of  the 
angular  velocity  will  remain  constant,  but  that  its'  axis  will  revolve 
uniformly,  and  will  be  always  perpendicular  to  the  axis  of  constant 
acceleration.  Now  the  axis  of  acceleration  is  always  horizontal 
when  the  rotating  body  is  a  top  which  spins  uniformly  with  its  axis 
of  revolution  inclined  to  the  vertical.  Hence  the  direction  of  the 
axis  of  the  top  will  rotate  uniformly,  and  will  be  always  perpendi- 
cular to  a  horizontal  line  (not  a  horizontal  plane)  which  rotates 
uniformly.  The  axis  must  therefore  revolve  at  a  uniform  rate 
around  the  vertical. 

Precession  of  the  equinoxes  is  due  to  angular  acceleration  of  the 
earth  about  an  equatorial  axis,  and  the  peculiar  motions  of  gyrostats 
have  a  similar  explanation. 

To  compound  two  angular  velocities  about  parallel  axes,  indicated 
in  magnitude  and  direction  by  AB  and  CD  (Fig.  40),  it  is  merely 
necessary  to  find  a  point  O  such  that  the  moments  of  AB  arid  CD 
about  it  are  equal  and  opposite  (§  49).  Let  p\,2hi  be  the  lengths 
of  perpendiculars  from  0  upon  AB  and  CD  respectively.  Since  the 
angular  velocity  of  0  due  to  rotation  about  AB  is  numerically  equal 

A  B 


to  AB,  it  follows  that  AB^  represents  the  velocity  of  0  perpendi- 
cular to  the  plane  of  the  paper ;  and,  similarly,  the  velocity  of  O 
perpendicular  to  that  plane  due  to  rotation  about  CD  is  CT>p.2 ;  and 
this  quantity  is  of  the  opposite  sign  to  the  former,  for  points  moving 


MOTION.  71 

along  AB  and  CD  revolve  oppositely  around  O.  Hence,  if  the 
areas  AOB  and  COD  are  equal,  the  point  O  is  at  rest ;  and  thus  the 
locus  of  0 — that  is,  a  line  through  0  parallel  to  AB  and  CD — is 
the  resultant  axis  of  rotation. 

Similar  reasoning  shows  that  the  resultant  axis  due  to  angular 
velocities  indicated  by  AB  and  DC  is  a  parallel  line  situated  above 
AB,  and  that  the  direction  of  rotation  coincides  with  that  of  AB, 
the  larger  of  the  two  components. 

To  find  the  effect  of  the  superposition  of  an  angular  velocity  w 
upon  a  linear  velocity  v,  or  of  a  linear  velocity  v  upon  an  angular 
velocity  w,  we  may  resolve  the  linear  velocity  into  its  two  compo- 
nents parallel  to  and  perpendicular  to  the  axis  of  rotation.  The 
effect  of  the  perpendicular  component  v'  is  to  shift  the  axis  parallel 
to  itself  through  a  distance  rf,  such  that  the  speed  dw  is  equal  and 
opposite  to  the  speed  v'.  The  parallel  component  simply  moves  the 
whole  body  in  the  direction  to  the  axis,  so  that  the  resultant  is  a 
twist. 

If  a  body  is  rotating  simultaneously  about  three  axes,  the  velocities 
being  representable  by  the  three  sides  of  a  triangle  taken  in  the  same 
direction  round,  the  effect  is  that  there  is  no  rotation ;  but  the  body 
is  translated  perpendicularly  to  the  plane  of  the  triangle  with  a  speed 
represented  by  twice  the  area  of  the  triangle.  If  one  of  the  rota- 
tions is  represented  by  a  side  of  the  triangle  taken  in  the  opposite 
direction  round,  there  is  no  translation ;  but  the  body  rotates  about 
an  axis  bisecting  the  two  sides  which  were  taken  in  the  same  way 
round.  The  rotation  round  this  axis  coincides  in  direction  with  the 
rotation  about  the  side  of  the  triangle  parallel  to  it,  and  the  angular 
velocity  is  twice  as  great  as  the  velocity  about  the  parallel  side. 

Corresponding  results  obtain  in  the  case  of  angular  velocities 
representable  by  the  sides  of  closed  plane  polygons. 

59.  Displacement  of  the  Parts  of  a  Non-Rig  id  Body. — Strain. — 
A  non-rigid  body  may  alter  in  form,  or  in  volume,  or  both.  Any  such 
definite  change  of  shape  or  bulk  is  called  a  Strain. 

Homogeneous  Strain. — When  all  parts  of  a  body,  originally 
similar  and  equal,  are  similarly  and  equally  strained,  the  strain  is 
said  to  be  homogeneous.  It  follows  that  parallel  straight  lines 
in  the  unstrained  body  become  parallel  straight  lines  in  the  strained 
body  ;  but,  in  general,  the  direction  of  the  lines  and  the  distance 
between  them  is  altered.  And  therefore  parallelograms  remain 
parallelograms,  parallelepipeds  remain  parallelepipeds,  and  any 
figure  or  surface  changes  into  a  similar  figure  or  surface.  Thus  a 
sphere  becomes  an  ellipsoid. 

Such  a  strain  is   completely  determined  when  we  know  what 


72  A    MANUAL    OF    PHYSICS. 

alteration  —  in  magnitude  and  direction — has  been  produced  in 
three  originally  non-coplanar  lines.  One  number  is  required  for 
each  line  to  fix  the  change  of  length  ;  and  two  numbers  are  required 
to  determine  the  change  of  direction  of  each  line.  In  all,  nine 
numbers  are  in  general  necessary. 

The  simplest  kind  of  homogeneous  strain  is  that  in  which  there  is 
uniform  expansion  or  compression  in  all  directions.  Any  line  in 
the  strained  figure  preserves  its  original  direction,  and  all  lines  are 
equally  altered  in  length.  Such  strain  occurs  in  the  compression  of 
fluids.  One  number  completely  determines  it. 

Next  in  order  of  simplicity  is  a  homogeneous  strain  in  which  lines 
in  one  definite  direction  are  unaffected  by  the  strain  ;  that  is,  the 
strain  is  c.onfined  to  planes  perpendicular  to  a  definite  direction,  and 
the  alteration  in  any  one  of  these  planes  is  precisely  similar  and 
equal  to  the  alteration  in  any  other.  It  is  usual,  therefore,  to  call 
such  a  distortion  a  plane  strain.  A  circle  in  the  unstrained  figure, 
drawn  in  one  of  the  planes  of  distortion,  becomes  an  ellipse  ;  and  so 
a  sphere  in  the  body  becomes  an  ellipsoid.  (In  the  cases  of  equal 
expansion  or  of  equal  contraction  in  all  directions  in  the  planes  of 
the  strain,  the  ellipsoid  is  an  oblate  or  a  prolate  spheroid  respec- 
tively. All  lines  which'  are  not  in  or  perpendicular  to  these  planes 
are  altered  in  direction.)  All  perpendicular  diameters  of  the  circle 
become  conjugate  diameters  of  the  ellipse ;  in  particular,  the 
principal  axes  of  the  ellipse  were  originally  perpendicular  diameters 
of  the  circle — from  which,  however,  they  usually  differ  in  direc- 
tion. Four  numbers  determine  a  plane  strain  of  the  most  general 
kind. 

When  the  principal  axes  of  the  ellipse  (called  the  strain-ellipse] 
into  which  the  circle  is  deformed  are  not  changed  from  their  original 
directions  in  the  unstrained  body,  the  strain  is  called  a  pure  (or 
non-rotational)  plane  strain.  In  this  case  the  distortion  consists  in 
extension  (or  contraction)  in  two  directions  at  right  angles  to  each 
other.  Any  rotational,  or  impure,  plane  strain  may  (so  far  as  the 
final  effect  is  concerned)  be  produced  by  a  pure  strain  superposed 
upon,  or  followed  by,  rotation  about  a  definite  axis  perpendicular  to 
the  plane.  In  a  pure  strain  every  line  except  a  principal  axis  has 
suffered  rotation  ;  and  it  follows  from  this  that  the  superposition  of 
two  pure  strains  generally  produces  an  impure  strain.  Hence  a 
body  may  be  distorted  by  three  plane  strains  in  succession,  and  yet 
(the  strains  being  properly  chosen)  be  left  unstrained,  but  rotated 
through  an  angle  about  a  definite  axis.  Three  numbers  completely 
characterise  a  pure  plane  strain. 

A  specially  important  case  is  that  in  which  there  is  no  alteration 


MOTION. 


73 


of  volume.  This  implies  elongation  in  one  direction  and  equal  con- 
traction in  another.  We  may  suppose  that  these  directions  are 
mutually  perpendicular,  giving  a  pure  strain  ;  for,  as  we  have  seen, 
any  other  strain  may  be  assumed  to  consist  in  a  pure  strain  followed 
by  rotation  as  of  a  rigid  body.  .  Let  oy  be  the  direction  of  the 
elongation,  and  let  ox  be  the  direction  of  contraction.  Let  abed  be 


y 


FIG.  41. 

a  rhombus  in  the  unstrained  figure,  which  becomes  the  rhombus 
a'b'c'd'  in  the  strained  state,  oa  being  equal  to  06',  oa'  being  equal  to 
06,  and  so  on.  We  have  then  ab  equal  to  a'b',  with  similar  results  for 
the  other  sides  of  the  rhombus.  It  is,  therefore,  obvious  that  there 
are  two  sets  of  planes  in  the  figure  which  experience  no  alteration, 
except  as  regards  position  ;  so  that  (rotation  excluded)  the  strain 

A'  I  ct  & 


FIG.  42. 

might  have  been  produced  by  holding  fast  one  plane  of  either  set — 
say  the  plane  through  cd  perpendicular  to  the  plane  of  the  paper — 
and  sliding  all  planes  parallel  to  it  through  a  distance  proportional 
to  their  distance  from  the  fixed  one,  until  the  originally  acute  angle 


74  A    MANUAL    OF    PHYSICS. 

of  the  rhombus  becomes  equal  to  its  supplement.  This  motion  is 
termed  shearing  motion,  and  the  strain  is  called  a  simple  shear. 

To  make  the  result  of  this  shear  coincide  with  the  result  of  the 
above  pure  strain,  we  must  turn  the  body  round  in  the  direction  of 
the  hands  of  a  watch  through  an  angle  b'db,  so  that  b'd  coincides 
with  bd — that  is,  through  an  angle  equal  to  half  the  difference 
between  the  obtuse  and  acute  angles  of  the  rhombus. 

In  the  most  general  homogeneous  strain,  a  sphere  in  the  un- 
strained body  becomes  an  ellipsoid  in  the  strained  state.  Any  set 
of  three  mutually  perpendicular  axes  become  mutually  conjugate 
diameters  of  the  ellipsoid.  In  particular,  the  three  principal  axes 
of  the  ellipsoid  (which  are  called  the  principal  axes  of  the  strain) 
were  originally  perpendicular  diameters  of  the  sphere.  These 
principal  axes  are  usually  rotated  from  their  initial  positions  ;  and, 
as  in  the  corresponding  case  of  plane  strain,  when  this  turning  of 
the  principal  axes  does  not  occur,  the  strain  is  said  to  be  pure  or 
non-rotational.  So  far  as  the  ultimate  result  is  concerned,  any 
impure  strain  may  be  looked  upon  as  due  to  a  pure  strain  followed 
by  a  rotation. 

Any  given  strain  may  be  produced  by  a  simple  shear  followed  by  an 
extension  (or  contraction)  perpendicular  to  the  plane  of  the  shear  which 
in  turn  is  succeeded  by  a  uniform  expansion  (or  compression).  For 
the  shear  may  be  continued  to  such  an  extent  as  to  give  the  proper 
ratio  of  the  maximum  and  minimum  axes  ;  and  the  perpendicular 
extension  will  then  give  the  proper  ratio  of  the  mean  axis  to  each 
of  the  other  two  ;  while,  lastly,  the  uniform  expansion  can  be  con- 
tinued to  such  an  extent  as  to  give  the  proper  magnitudes  of  the  axes. 

60.  Non-Homogeneous  Strain. — So  long  as  rupture  does  not 
occur,  all  displacements  in  a  portion  of  matter  are  essentially  con- 
tinuous. Hence,  however  greatly  the  displacements  may  change 
throughout  the  body,  we  can  always  consider  a  portion  so  small 
that,  within  its  limits,  the  strain  is  homogeneous. 

Let  P  and  Q  be  two  near  points — so  near  that  the  strain  is 
homogeneous,  and  let  8x,  dy,  <te,  or  |,  rj,  £,  be  the  co-ordinates  of  Q 
relative  to  P.  If  u,  v,  iv,  be  the  components  of  the  displacement 
of  P  parallel  to  the  axes  of  x,  y,  and  z,  respectively,  we  may  denote 
by  u  +  du,  v  +  dv,  w+ dw  the  corresponding  quantities  for  Q,  so 
that  du,  dv,  dw  are  the  components  of  the  relative  displacement  of 
P  and  Q.  Each  of  these  components  will  in  general  depend  upon 
the  values  of  £,  ?;,  and  £  ;  and  each  must  be  a  linear  function 
(£  29)  of  these  quantities  since,  in  homogeneous  strain,  straight  lines 
remain  straight  lines.  Now,  du/dx  being  the  #-rate  of  variation  of 
u,  du/dx  .  dx  is  the  change  of  u  due  to  the  change  Bx,  and  so  on  ; 


*       MOTION.  75 

so  that  dii  =  du/dx  .  d<£-\-du/dy  .  cy  -\-dwfdz  .  Sz.     But,  instead  of  du, 
we  may  write  the  equivalent  quantity  d%,  and  so  finally 

(hi       du       du 


-_  . 

dx       dy       dz 

The  quantities  u,  v,  and  w  being  known  functions  of  x,  y,  and  z, 
these  equations  enable  us  to  determine  fully  the  nature  of  the 
strain  in  the  neighbourhood  of  any  given  point. 

The  multipliers  of  £,  »;,  and  £,  in  these  equations,  are  the  nine 
numbers  which  determine  the  strain  (§  59). 

61.  Motion  of  Fluids.  —  While  the  parts  of  a  rigid  body  cannot 
suffer  relative  displacement,  the  parts  of  a  non-rigid  solid  can  be  dis- 
placed relatively  ;  but  the  magnitude  of  any  displacement  is  not  un- 
limited, for,  if  it  be  too  large,  the  body  will  be  ruptured.  In  an 
infinite  expanse  of  fluid  there  is  no  limit  to  the  possible  increase 
of  distance  between  two  originally  near  parts. 

A  Line  of  Flow  in  a  moving  fluid  is  defined  as  a  line  so  drawn 
that  its  direction  at  any  point  coincides  with  the  direction  of  motion 
of  the  fluid  at  that  point.  It  may,  or  it  may  not,  be  the  actual 
path  of  any  particle  of  the  fluid.  In  illustration  of  this  we  may 
consider  the  motion  of  points  in  a  spinning  top  (^  58).  At  any 
instant  the  line  of  flow  of  a  given  point  is  a  circle  drawn  round  the 
axis  of  the  top.  But  the  axis  is  itself  in  motion,  so  that  the  path 
of  the  point  merely  coincides  with  the  line  of  flow  for  an  indefinitely 
small  distance.  Similarly,  when  the  lines  of  flow  in  a  fluid  are  in 
motion,  the  path  of  any  particle  only  coincides  with  one  line  of 
flow  for  an  indefinitely  small  interval  of  time. 

When  the  lines  of  flow  are  fixed,  so  that  they  are  actual  paths  of 
particles,  the  motion  is  said  to  be  steady,  and  the  lines  are  called 
stream-lines. 

If  lines  of  flow  are  drawn  through  all  points  of  a  closed  curve,  a 
Tube  of  Flow  is  formed.  None  of  the  fluid  inside  such  a  tube  ever 
passes  out  of  it,  and  none  ever  enters  it  from  the  outside. 

Whatever  be  the  nature  of  the  strain  throughout  a  solid,  we  may 
consider,  instead  of  the  total  strain,  the  strain  produced  in  a 
given  indefinitely  small  period  of  time.  The  displacements  in 
that  period  are  evidently  proportional  to  the  instantaneous  velocities 


76  A    MANUAL    OF    PHYSICS. 

of  the  various  parts.  And,  hence,  all  the  results  which  we  have 
obtained  regarding  displacements  in  a  non-rigid  solid  have  a  direct 
application  in  the  discussion  of  fluid  motion. 

Just  as  rotational  strain  may  exist  in  a  solid,  so,  in  a  fluid,  there 
may  be  rotational,  or,  as  it  is  termed,  Vortex  Motion.  A  line, 
drawn  in  the  fluid  so  that  its  direction  coincides  at  any  point  with 
the  direction  of  the  axis  of  rotation  at  that  point,  is  called  a  vortex 
line.  And  a  tube  formed  by  vortex  lines  drawn  through  all  points 
of  an  infinitely  small  closed  curve  forms  a  vortex  tube,  and  is  said  to 
enclose  a  vortex  filament.  In  most  cases  of  fluid  motion  in  which 
vortices  exist,  the  vertically  moving  parts  occupy  only  a  small  pro- 
portion of  the  whole  volume  of  the  fluid. 


FIG.  43. 

If  ds  represents  an  infmitesimally  small  length  of  a  curve  drawn 
between  any  two  points  in  a  moving  liquid,  while  v  represents  the 
velocity  parallel  to  the  curve  at  any  point,  the  integral  of  the 
quantity  vds  is  called  the  Circulation  along  the  curve.  If  we  sur- 
round the  curve  by  a  tube  (Fig.  43),  the  sectional  area  of  which  is 
numerically  equal  to  the  speed  along  the  curve,  the  volume  of  that 
tube  is  numerically  equal  to  the  circulation  along  the  curve.  The 
positive  or  negative  sign  must  be  attached  according  as  the  circulation 
is  in  the  positive  or  negative  direction  round  the  curve.  When  the 
curve  is  closed  and  lies  in  a  plane,  we  may  speak  of  the  circulation 
'  round  the  enclosed  area.'  It  is  evident  that  the  circulation  round 
any  area  is  equal  to  the  sum  of  the  circulations  round  its  parts, 
for  the  circulations  round  a  common  boundary  are  equal  and  of 
opposite  sign. 

Shearing  Motion. — The  results  which  we  have  obtained  regarding 
pure  homogeneous  plane  strains  apply  directly  to  this  case  of  fluid 
motion.  And  it  is  easy,  in  addition,  to  deduce  useful  results  regard- 
ing the  circulation  of  the  fluid. 

(1)  The  circulation  along  any  two  plane  conterminous  curves, 
which  nowhere  lie  at  a  finite  distance  from  each  other,  is  the  same. 


MOTION.      .  77 

To  see  this,  let  abc  and  ac  be  portions  of  two  such  curves,  and  let 
the  straight  line  mn  indicate  the  velocity  v,  which  is  constant  all 
along  these  portions  provided  they  are  small  enough.  Hence  the 
circulations  round  abc  and  ac  are  the  products  of  v  into  the  projec- 
tions of  abc  and  ac  respectively  upon  mn.  But  these  projections 
are  equal,  which  proves  the  result. 


FIG.  44. 

(2)  The  circulations  round  any  two  similarly  situated  and  equal 
plane  areas  are  equal.     For  the  velocities  in  the  one  curve  relatively 
to  the  point  o  are  equal  to  those  in  the  other  curve  relatively  to  the 
corresponding  point   o'.      Hence   the   circulation   round   s'   differs 
from  the  circulation  round  s  by  the  product  of  the  relative  speed 
of  o'  and  o  into  the  projection  of  s'  upon  the  line  of  relative  motion 
of  o  and  o'.     But  this  vanishes,  since  s'  is  a  closed  curve. 

(3)  The  circulation  round  any  plane  curve  is  proportional  to  its 
area.     For  we  may  divide  the  area  into  a  series  of  indefinitely 
small,  similarly  situated,  and  equal,  parallelograms.     The  circulation 


FIG.  46. 

round  each  of  these  is  equal  by  (2).  And  now,  since  the  edge  of  the 
area  formed  by  these  parallelograms  is  nowhere  finitely  apart  from 
the  given  curve,  the  result  follows  from  (1). 

(4)  The  circulation  round  any  plane  area  is  equal  to  twice  the 
area  multiplied  by  the  angular  velocity  of  the  fluid  round  a  perpen- 
dicular axis.  Let  the  area  be  an  indefinitely  small  circular  area. 
From  the  analogy  to  strain,  we  see  that  any  instantaneous  motion 
may  be  broken  up  into  a  pure  part  and  a  rotational  part.  The 


78  A    MANUAL    OF    PHYSICS., 

motion  corresponding  to  the  pure  part  is  constant  over  the  small 
area,  and  therefore  contributes  nothing  to  the  circulation.  The 
rotational  part  gives  a  tangential  speed  wr,  where  w  is  tjie  angular 
velocity  round  a  perpendicular  axis  through  the  centre,  and  r  is 
the  radius  of  the  circle.  Hence  the  circulation  is  2:rr  .  rw  =  27rrsw. 
But  Trf2  is  the  area  of  the  circle,  and  so  the  proposition  is  true  in  this 
case ;  and,  w  being  constant,  we  see,  by  (3),  that  it  holds  in  all 
cases. 

Heterogeneous  Motion. — A  portion  of  the  fluid  may  be  taken  so 
small  that  the  motion  is  homogeneous  throughout.  The  previous 
results  are  then  applicable. 

Keasoning  similar  to  the  above  shows  that  the  circulation  round 
any  closed  curve  (plane  or  not)  in  the  fluid  is  equal  to  twice  the 
integral  of  the  normal  angular  velocity  over  any  surface  bounded 
by  the  curve.  For  a  sufficiently  small  portion  of  the  surface  may 
be  assumed  to  be  plane,  and  the  result  (4)  applies.  Hence,  if  a 
closed  surface  be  taken,  and  the  normals  at  all  points  be  drawn 
outwards  alone,  or  inwards  alone,  the  integral  of  the  angular 
velocity  over  the  surface  is  zero.  For  if  we  draw  any  closed  curve 
round  the  closed  surface,  the  integral  over  each  portion  of  the  surface 
corresponds  to  equal  but  opposite  circulation  round  the  closed  curve. 

Vortex  Motion. — The  above  conclusion  may  be  applied  to  the  case 
of  a  finite  portion  of  a  vortex-tube.  The  sides  of  such  a  tube  are  parallel 
to  the  axes  of  rotation.  Therefore  the  ends  only  contribute  to  the 
integral  of  the  angular  velocity ;  and  so  the  integral  for  each  end 
must  be  equal  in  magnitude,  and  it  will  be  of  the  same  sign  in  both 
cases  if  the  normals  are  drawn  in  the  same  direction  along  the  tube. 
Hence  the  circulation  is  the  same  at  all  sections  of  a  vortex-tube. 
The  tube  being  small,  the  angular  velocity  is  therefore  inversely 
proportional  to  the  cross-section.  Hence  the  vortex  rotates  faster 
the  thinner  it  is. 

We  conclude  also  that  a  vortex  must  either  return  into  itself, 
forming  a  closed  circuit,  or  that  its  ends  must  be  at  the  surface  of  the 
liquid  ;  for  the  velocity  of  rotation  can  neither  abruptly  change 
nor  become  infinite  within  the  liquid.  A  smoke-ring  exemplifies 
the  former  case;  the  eddies  formed  round  the  edge  of  the  hand, 
when  it  is  dipped  into  water  and  drawn  rapidly  along,  illustrate  the 
latter. 


CHAPTER  VI. 

MATTER     IN     MOTION. 

62.  Force. — The  fundamental  property  of  matter,  which  distinguishes 
it  from  the  only  other  real  thing  in  the  universe,  is  inertia.  And, 
in  consequence  of  inertia,  when  we  move  a  body  we  are  conscious 
of  making  some  exertion,  and  are  accustomed  to  say  that  we  exert 
force.  Hence  it  is  usual,  as  Newton  did,  to  speak  of  force  as  the 
cause  of  motion ;  and  the  force  may  be  of  the  nature  of  a  push,  a 
pull,  an  attraction,  a  repulsion,  etc. 

We  do  not  yet  know  the  nature  of  the  physical  process  going  on  in 
matter  which  is  in  a  state  of  tension,  and  so  we  figure  it  to  ourselves 
by  means  of  the  mental  impression  caused  by  the  muscular  sense. 
But  loudness  and  brightness,  though  they,  as  we  shall  see  later,  are 
mere  subjective  impressions,  yet  correspond  to  certain  physical 
realities.  We  may  therefore  proceed  to  inquire  whether  or  not 
there  is  some  physical  process  corresponding  to  the  impression  of 
force. 

We  have  already  obtained  (§  7)  a  kinetic  measure  of  energy ; 
and  by  means  of  the  new  idea  of  force,  we  can  now  deduce  a 
statical  measure  of  it.  Work  is  done  when  we  move  a  body  against 
the  action  of  a  force  which  we  assert  to  be  the  cause  of  motion  in 
the  opposite  direction.  If  the  force  is  constant,  we  know  that  the 
work  which  is  done  is  proportional  to  the  distance  through  which 
the  body  is  moved  against  the  action  of  the  force,  for  every  equal 
addition  to  the  distance  is  made  under  precisely  similar  circum- 
stances. Also  we  know  that  the  amount  of  work  which  is  done  is 
proportional  to  the  force,  it  being  more  and  more  difficult  to  produce 
the  displacement  according  as  the  opposing  force  is  greater.  Hence, 
provided  we  define  the  unit  of  work  as  the  work  done  by  unit  force 
acting  through  unit  distance,  we  may  write 

w  =fs, 
where  w,  /,  and  s  represent  respectively  the  work,  the  force,  and 


80  A    MANUAL    OF    PHYSICS. 

the  distance.  But,  by  the  principle  of  conservation  of  energy,  if  a 
mass  m  is  projected  with  speed  v,  and  is  brought  to  rest  by  the  action 
of  the  force,  we  know  that  the  work  is  equivalent  to  the  kinetic 
energy  which  is  lost.  Therefore 

±mv2=fs. 
From  this  equation,  denoting  the  energy  by  e,  we  deduce 

de/ds=f. 
That  is,  force  is  the  space-rate  of  variation  of  energy. 

Force  is  not  conserved  as  energy  is,  although  it  ma}"  possess  that 
kind  of  conservation  spoken  of  in  §  12.  Indeed,  Newton's  third 
law  of  motion  asserts  that  the  total  algebraic  sum  of  the  forces  in 
the  universe  is  zero. 

And  now,  having  arrived  at  a  clear  understanding  of  what  '  force  ' 
really  is,  we  may  use  the  word,  or  any  of  its  special  equivalents,  in 
subsequent  sections,  and  speak  of  force  as  the  cause  of  motion, 
without  producing  confusion  of  ideas. 

There  is  no  such  thing  in  nature  as  a  material  point.  How- 
ever small  a  particle  of  matter  may  be,  it  always  has  a  finite 
surface,  and  occupies  a  certain  volume.  And  no  actual  force  acts 
at  a  point  merely  ;  it  is  distributed  throughout  a  volume,  or  is 
applied  over  a  surface.  As  an  example  of  the  former  class  of  forces, 
we  may  take  the  force  of  gravitation  ;  as  an  example  of  the  latter 
class,  we  may  take  the  force  of  friction,  i.e.,  the  tangential  force 
which  resists  the  sliding  of  portions  of  matter  over  each  other. 
[This  tangential  force  is  independent  of  the  area  of  the  surface 
of  contact  of  the  two  bodies  —  so  long,  at  least,  as  the  surface 
of  contact  is  not  so  small  that  sliding  motion  cannot  occur  with- 
out producing  abrasion  of  the  substances.  It  is  in  general  pro- 
portional to  the  normal  pressure  between  the  bodies  ;  but,  in  many 
substances,  it  depends  greatly  upon  the  time  during  which  the 
contact  has  lasted.  It  may  be  much  reduced  by  the  use  of  proper 
lubricants. 

The  law  of  friction  may  be  expressed  by  the  equation 

F  =  /B, 

where  F  is  the  force  of  friction,  R  is  the  normal  pressure,  and  /<'  is 
a  constant,  called  the  co-efficient  of  kinetic  friction. 

When  the  forces  which  act  so  as  to  produce  motion  are  just  in- 
sufficient to  overcome  friction,  this  equation  becomes 


where  p,  —  called  the  '  co-efficient  of  statical  friction  '  —  is,  as  experi- 
ment shows,  a  constant  of  greater  numerical  magnitude  than  /w'. 


MATTER   IN    MOTION.  81 

It  follows  that  sliding  motion  will  continue  under  the  action  of 
forces  which  were  inadequate  to  start  the  motion.] 

63.  The  Laws   of  Motion.  —  Newton's   three   Laws  of   Motion 
(expressed  in  terms  of  force  regarded  as  the  cause  of  motion)  form 
at  present  the  simplest  foundation  for  the  study  of  the  phenomena 
of  moving  matter.     The  science  which  deals  with  these  phenomena 
has,  therefore,  been  called  Dynamics,  i.e.,  the  science  which  treats 
of  the  action  of  force  upon  matter.     It  is  usual,  also,  to  divide  the 
subject  into  two  parts — Kinetics  and  Statics — according  as  motion 
is,  or  is  not,  produced.     It  is  impossible  to  doubt  that  ultimately  a 
more  fundamental,  and  at  least  equally  simple,  basis  will  be  obtained 
in  connection  with  the  principles  of  energy.     We  can  even  at  present 
make  such  a  substitution  for  Newton's  Laws,  barring  the  simplicity. 

64.  The  First  and  Second  Laws. — The  First  Law  asserts  that  evert/ 
body  maintains  its  state  of  rest,  or  of  uniform  motion  in  a  straight 
line,  except  in  so  far  as  it  is  caused  by  force  to  alter  that  state. 

The  '  rest '  here  referred  to  is,  of  course,  relative  rest.  And, 
instead  of  the  last  clause,  we  might  say  '  so  long  as  it  remains  in  a 
region  of  constant  potential.'  This  law,  in  fact,  asserts  the  conserva- 
tion of  the  energy  of  the  particular  body  considered,  potential  energy 
being  regarded  (§  8)  as  energy  which  has  passed  to  a  connected  system. 

Uniformity  of  motion  in  magnitude  and  direction  does  not  enable 
us  to  say  that  no  force  is  acting  upon  the  body,  but  only  that  the 
resultant  of  all  the  forces  is  zero — that  they  can  be  combined  into 
two  equal  and  opposite  forces.  This  means  that  the  body  may  be 
simultaneously  gaining  and  losing  energy  at  precisely  equal  rates. 

The  law  implies,  also,  that  force  must  be  acting  upon  a  body  if  the 
direction  of  its  motion  alters,  the  magnitude  remaining  constant. 
[In  this  case  the  body  moves  along  an  equipotential  surface.] 

The  Second  Law  gives  the  relation  between  force  and  the  effect 
which  it  produces.  It  states  that  change  of  motion  is  proportional 
to  force,  and  is  in  the  direction  in  ivhich  the  force  acts.  By 
motion  Newton  meant  what  is  now  called  momentum — the  product 
of  the  mass  and  the  velocity  of  the  moving  body.  Of  course,  the 
change  of  momentum  is  proportional  to  the  time  during  which  a 
constant  force  has  acted,  so  that  we  may  express  the  second  law  by 
the  equation 

ft  —  mv, 

v  being  the  change  of  speed  produced  during  time  t  in  the  given 
mass  in  by  the  average  force  /.  The  actual  force  at  any  instant  is 
got  by  making  the  time  indefinitely  small,  when  the  equation  becomes 

/=  ma, 

6 


82  i  A    MANUAL    OF    PHYSICS. 

«  being  the  acceleration.  And  this  form  of  the  equation  may  be 
used  in  all  cases,  whatever  be  the  value  of  /,  provided  that  /  and  « 
represent  the  average  values  of  the  force  and  the  acceleration  during 
the  time  t.  Both  equations  involve  the  assumption  (or,  rather,  the 
definition)  that  unit  force  is  the  force  which,  acting  upon  unit  mass  for 
unit  time,  produces  unit  change  of  speed.  [From  the  latter  we  get 
fds  =  mavdi  =  mvvdt  =  mvdv,  and  therefore/s  =  ^mv~,  so  that  our  new 
definition  of  force  is  equivalent  to  the  former  one  (}J  62).] 

If  two  forces,  /i  and  /2,  act  upon  equal  masses,  the  accelerations 
produced  are  proportional  to  those  forces,  for  /a//o  =  a1/a.2.  Thus  the 
Second  Law  of  motion  gives  us  a  method  of  comparing  forces.  It 
also  enables  us  to  compare  masses.  For,  if  two  equal  forces  act 
upon  different  masses,  in^  and  m2,  we  have  mla^=m./i.2  ;  i.e.,  the 
accelerations  produced  are  inversely  proportional  to  the  masses. 

By  means  of  the  first  two  laws  alone  we  can  investigate  the 
motion  of  a  material  point  or  of  a  set  of  disconnected  particles. 

For  example,  in  the  equations  of  motion  given  in  £  42,  we 
have  only  to  introduce  the  mass  m  as  a  factor  on  each  side  in  order 
to  obtain  various  dynamical  quantities.  Thus  the  equation  x  =  g 
becomes  mx  =  mg.  The  quantity  mg  thus  represents  the  force  with 
which  the  earth  is  attracting  the  mass  m  —  i.e.,  the  weight  of  the 
body.  Hence,  if  m  be  the  weight,  we  have  the  equation 


which  expresses  the  fundamental  distinction  between  Weight  and 
mass.  The  mass  m  is  fixed  in  amount  ;  the  weight  w  varies  when 
g  varies,  and  might  be  caused  to  vanish  by  taking  the  body  to  a 
region  where  g  was  zero. 

It  is  frequently  convenient  to  use,  instead  of  m,  the  equivalent 
quantity  V(>,  where  V  is  the  volume  of  the  mass  m  and  p  is  the  mass 
per  unit  volume,  which  is  called  the  density  of  the  substance. 

Again,  the  equation  x  =  gt  becomes  mx  —  ingt.  But  'nix  is  the 
momentum  produced  in  the  body,  which  we  thus  see  to  be  propor- 
tional to  the  time  during  which  the  body  has  been  falling  under  the 
action  of  gravity. 

Also,  instead  of  x-=-  V-  —  2^.r,  we  have  fynx2  =  fyn'V-—ingx,  or 
•jra(V2  —  a;2)  =  mgx,  which  tells  us  that  the  loss  of  kinetic  energy  is 
proportional  to  the  distance  through  which  the  body  has  risen. 

In  £  43  it  was  shown  that  the  acceleration  of  a  point  moving  in  a 
circle  of  radius  r  with  uniform  speed  v  is  v'2/r  towards  the  centre. 
If  the  point  have  mass  m,  this  corresponds  to  a  central  force  mv-/r, 
which,  e.g.,  in  the  case  of  a  stone  revolving  in  a  sling,  is  supplied 


MATTER    IN    MOTION.  83 

by  the  tension  of  the  cord.  The  necessity  for  this  central  force 
gave  rise  to  the  erroneous  idea  of  a  '  centrifugal  force,'  which  it  was 
supposed  to  balance.  In  accordance  with  the  First  Law,  the  body 
tends  to  move  along  a  tangent,  and  not  from  the  centre,  and  the 
apparent  force  is  really  a  result  of  inertia. 

65.  Further  Discussion  of  the  Second  Law. — The  above  examples 
involve  the  application  of  a  single  force,  constant  in  magnitude  and 
direction,  to  a  material  particle.     But  the  Second  Law  enables  us 
also  to  investigate  the  motion  of  a  material  particle  under  the  action 
of   any  number  of   forces  acting  simultaneously,  for  it  implicitly 
asserts  that  each  -force  acts  independently  of   all  the  others,  i.e., 
the  effect  produced  by  any  force  is  the  same  as  it  would  be  if  that 
force  alone  acted  upon  the  particle  when  at  rest. 

To  completely  specify  a  force  we  require  to  know  its  magnitude, 
the  direction  in  which  it  acts,  and  the  place  at  which  it  is  applied. 
Hence  a  force  is  a  vector  quantity,  and  so  the  resultant  of  any 
number  of  forces  acting  simultaneously  upon  a  material  point  is  to 
be  found  by  the  ordinary  law  for  the  composition  of  vectors.  Indeed 
this  follows  at  once  from  the  Second  Law,  since  the  forces  are  pro- 
portional to  the  accelerations  which  they  produce. 

Hence,  when  a  particle  is  acted  upon  by  any  number  of  forces, 
we  need  only  consider  it  as  moving  under  the  action  of  the  single 
resultant  force. 

Since  a  mere  particle  has  only  three  degrees  of  freedom — all  trans- 
lational — three  conditions  completely  determine  its  motion.  If  X,  Y,  Z 
be  the  components  of  the  resultant  force  in  the  directions  of  the  axes 
of  x,  y,  z,  respectively,  these  conditions  are,  by  the  Second  Law, 
ma;  =  X;  mv/^Y;  mz  =  Z. 

In  particular,  the  conditions  of  equilibrium  are 
X  =  0;  Y  =  0;  Z  =  0.- 

66.  Special  Examples.  —  We  shall  now  apply  these   results  to 
some  special  cases  of  motion  of  a  material  particle. 

(1.)  A  particle  of  mass  m  is  projected  with  initial  speed  va.  It 
experiences  a  resistance  which  is  proportional  to  its  velocity.  In- 
vestigate the  motion.  • 

We  may  suppose  that  the  motion  is  in  the  direction  of  the  axis  of 
«c,  so  that  we  have  only  to  consider  the  single  equation 

mx  =  -  kx, 
where  k  is  constant.     Multiplying  each  side  by  dt,  this  becomes 

mdx  =  —  kdx. 
The  integral  is  mx  =  c-7cx. 

6—2 


84  A    MANUAL    OF    PHYSICS. 

To  determine  the  value  of  the  constant  c,  we  observe  that  v0  is  the 
(given)  value  of  x  when  x  =  o.     Hence  c  —  mv0.     Therefore,  finally, 

m(va—  x)  =  kx. 

This  result  shows  that  the  particle  will  come  to  rest  at  a  distance 
mva/k  from  the  point  of  projection. 

We  may  write  the  equation  in  the  form 
mdx 


mvu  -  kx 

This  gives  (§  38)  t  =  TF0—m/k  log  (mv0-kx),  and  so,  since  t  =  o  when 
<£  =  o,  we  have  T0  =  m/k  log  mva.  But  when  £  =  TU  we  get  x—ntv0/k. 
That  is  rr,,  —  mlk  log  mvtt  is  the  value  of  the  time  which  elapses  until 
the  particle  comes  to  rest. 

(2.)  A  particle  slides  from  rest  down  an  inclined  plane  under  the 
action  of  gravity.  How  long  wrill  it  take  to  move  over  a  given  dis- 
tance, and  what  will  be  its  speed  of  motion  when  it  reaches  the 
given  point  ? 

Let  m,  «,  E,  and  F  represent  respectively  the  mass  of  the  particle, 
the  inclination  of  the  plane  to  the  horizon,  the  normal  pressure, 
and  the  force  of  friction.  We  may  take  the  axis  of  x  in  the  direc- 
tion of  motion,  so  that  we  get 

mx  =  mg  sin  a  —  F. 
If  /*'  is  the  co-efficient  of  friction,  this  becomes 

mx  =  mg  sin  a  -  jt*'R. 

If  the  axis  of  y  be  taken  perpendicular  to  the  plane,  the  other 
equation  is 

wy  =  0  =  R  —  mg  cos  a. 
Hence 

x  =  ff  (sin  n  -  ju'  cos  a)  =  A  (say). 


{tru 

-of, 


FIG.  47. 

This  quantity  is  independent  of  the  mass  of  the  particle.     Multi- 
plying by  dt  we  get  dx  =  A.dt,  from  which 


MATTER   IN    MOTION.  85 

where  v0  is  the  initial  speed,  and  therefore  is  zero  by  the  given 
conditions.     Hence,  again, 


If  we  take  the  origin  at  the  point  from  which  the  particle  starts, 
we  get  a?u  =  0,  so  that  the  time  taken  to  move  over  the  distance 


(sn  a  -  //  cos   a) 
And  the  speed  attained  is 


(sin  a  —  fi'  cos  a). 
The  kinetic  energy  gained  by  the  particle  is  therefore 
mgx  (sin  a  —  n'  cos  a). 

(3.)  A  material  particle  is  attached  by  an  elastic  cord  to  a  point 
on  an  inclined  plane  down  which  it  would  slide  under  the  action  of 
gravity  if  not  so  attached.  Find  the  limiting  values  of  the  tension 
in  the  cord  between  which  motion  will  not  occur. 

T  being  the  tension  in  the  cord,  and  the  other  quantities  having 
the  same  meaning  as  in  (2),  we  get 

mx=Q=mg  sin  «—  T±[img  cos  «. 

From  this  equation  the  two  values  of  T  may  be  found.  The 
sign  -|-  corresponds  to  the  case  in  which  the  cord  has  its  greatest 
extension  so  that  friction  acts  down  the  plane.  The  sign  —  indicates 
that  the  particle  is  just  on  the  point  of  sliding  down,  T  having  its 
smallest  possible  value. 

(4.)  A  particle  of  mass  in  is  swung  round  in  a  vertical  circle  by 
means  of  a  cord  of  length  1.  What  must  be  its  angular  velocity  in 
order  that  the  string  may  just  be  slack  when  the  particle  is  at  the 
highest  point  of  its  path  ? 

The  downward  force  is  the  weight  of  the  particle,  which  must 
be  balanced  by  the  reaction  to  acceleration  (the  so-called  '  centri- 
fugal force  ').  Hence,  at  the  highest  point, 

g  =  ^l, 

where  w  is  the  angular  velocity. 
At  the  lowest  point  we  have 


67.  Dynamical  Similarity.  —  We  may  write  the   expression  for 
Newton's  Second  Law  (§  64)  in  the  form 


8£  A    MANUAL    OF    PHYSICS.     f 

where  I  is  the  distance  which  the  mass  m  moves  over  from  rest,  in 
the  time  t,  under  the  action  of  the  force  /.  And  we  may  further 
regard  this  equation  as  a  dimensional  equation  ($  27)  ;  in  which 
case  the  sign  of  equality  merely  means  that  the  dimensions  of  the 
quantity  on  the  left-hand  side  of  the  equation  are  identical  with 
those  of  the  expression  on  the  right-hand  side  of  the  equation. 
But,  from  a  dimensional  equation,  we  cannot  make  any  deduction 
regarding  the  absolute  magnitude  of  any  of  the  quantities  which  are 
involved,  for  the  equation  simply  asserts  proportionality  of  magni- 
tude between  its  various  terms.  Still,  by  a  suitable  definition  of 
units,  we  can  pass  from  the  dimensional  to  the  ordinary  equation. 
Thus,  in  the  above  equation,  we  may  define  unit  force  as  the  force 
which,  acting  on  unit  mass  for  unit  time,  causes  the  unit  of  mass  to 
move  over  unit  distance  from  rest  ;  or  we  might  adopt  the  definition 
of  §  64. 

The  idea  of  dimensions  is  of  great  importance  in  physics.  It 
affords  a  useful  check  on  the  accuracy  of  algebraical  work  ;  for  the 
dimensions  of  all  the  terms  in  a  physical  equation  must  be  the 
same.  But  its  use  is  not  limited  to  this  extent.  For  example,  we 
may  write  the  equation 


in  the  form 


from  which  we  see  that  if,  in  two  similar  material  systems,  the 
forces,  masses,  and  lengths,  are  in  the  ratios  a/1,  /3/1,  and  y/1, 
respectively,  and  if  the  systems  begin  to  move  in  precisely  similar 
manners,  the  motions  ivill  continue  to  be  similar,  provided  that  we 
corn-pare  them  after  the  lapse  of  intervals  of  time  which  are  in  the 

ratio  of   \f  —  to  unity  in  the  two  systems. 

This  principle,  which  was  proved  otherwise  by  Newton,  has  been 
called  the  Principle  of  Dynamical  Similarity.  Later  on,  we  shall 
get  various  examples  of  its  use.  (See  ^  73,  76,  124,  159.) 

68.  The  Third  Law.  —  Hitherto  we  have  not  discussed  the  motion 
of  portions  of  matter  between  which  there  is  mutual  action  of  any 
kind.  The  first  two  laws  of  motion  do  not  enable  us  to  solve  such 
problems.  The  requisite  additional  information  is  given  by  the  Third 
Law  of  Motion  :  The  mutual  actions  between  any  two  bodies  are 
equal  and  oppositely  directed. 


MATTER    IN    MOTION. 


87 


A  stress  is  defined  as  a  system  of  equilibrating  forces,  and  so  we 
may  put  the  above  law  into  the  form  :  The  mutual  action  between 
any  two  bodies  is  of  the  nature  of  a  stress. 

No  one  will  question  the  truth  of  this  law  in  the  cases  in  which 
the  various  masses  concerned  are  in  equilibrium.  Thus,  when  a 
book  lies  upon  a  table,  we  say  that  the  table  reacts  upon  the  book 

j±-rn  CF|  F, 


FIG.  48. 

with  a  pressure  which  is  equal  to  its  weight.  But  it  is  by  no  means 
so  evident  that  a  body,  when'  pulled  along  by  means  of  a  cord, 
pulls  backwards  with  a  force  which  is  precisely  equal  to  that  by 
which  it  is  dragged  forwards.  In  order  to  see  how  this  can  be  we 
must  consider  all  the  forces  which  are  acting  upon  the  moving  body. 
Let  B,  Fig.  48,  be  the  body  and  let  F  be  the  force  with  which  it  is 
pulled  in  the  direction  of  F'F.  Also  let  F"  be  some  other  force 
acting  upon  B  in  the  opposite  direction. 

The  equilibrium  of  B  is  determined  solely  by  the  equality  of  the 
forces  acting  upon  it,  i.e.,  of  the  forces  F  and  F",  and  is  not  at  all 
affected  by  the  force  F'  with  which  B  reacts  upon  the  pulling  body. 
Hence  the  equality  of  the  forces  F'  and  F  is  a  matter  which  is 
entirely  independent  of  the  equality  of  F"  and  F,  and  can  only  be 
proved  by  experiment.  It  is  needless  to  add  that  all  Newton's 
laws  express  the  results  of  experiment  or  of  observation. 

But  (as  Newton  himself  pointed  out)  we  may  regard  '  action,'  not 
merely  as  force  but,  as  the  product  of  force  into  the  speed  which  it 
produces  in  the  body  upon  which  it  acts.  Now,  the  speed  produced 
being  the  (time)  rate  at  which  the  force  moves  the  body,  this  pro- 
duct is  (§  62)  the  (time)  rate  at  which  work  is  done  by  the  force. 
[In  modern  terminology  this  is  called  the  Activity.]  Hence  a 
second  interpretation  of  the  third  law  is  that  the  (time)  rate  at 
which  a  set  of  forces  do  work  upon  a  given  system  is  equal  and 
opposite  to  the  rate  at  'which  the  reacting  forces  do  ivork.  Had 
Newton  been  aware  that  heat  was  a  form  of  energy  this  would  (see 
Thomson  and  Tait's  Elements  of  Natural  Philoso2)hy)  have  been 
a  complete  statement  of  the  modern  principle  of  conservation  of 
energy ;  but,  in  his  day,  it  was  supposed  that  work  spent  in  over- 
coming friction  is  unavoidably  and  entirely  lost. 

Taken  in  conjunction  with  the  Second  Law,  this  law  enables  us 
to  investigate  the  motion  of  bodies  which  impinge  upon  each  other. 


88  A    MANUAL    OF    PHYSICS. 

To  avoid  unnecessary  complications  we  may  assume  that  two 
smooth  spheres,  of  masses  mx  and  m.2  respectively,  are  moving  in 
the  direction  of  the  line  joining  their  centres  with  speeds  vl  and  v.2 
respectively,  and  that  after  impact  their  velocities  are  v\  and  i>'2. 
In  most  practical  cases  the  time  of  impact  is  a  very  small  fraction  of 
a  second,  and  the  force  is  very  large,  so  that  it  is  impossible,  without 
special  appliances,  to  determine  the  values  of  these  quantities.  But 
the  value  of  their  product,  called  the  Impulse,  can  generally  be 
found  without  much  difficulty.  The  third  law  tells  us  that  the  im- 
pulse is  the  same  for  each  body,  and  hence 

7>?1(v'1  —  Vj)  =  m.2(v»  —  v'.2). 

In  addition  to  this  we  have  the  condition,  determined  experimentally 
by  Newton, 

v\  —  v'.,  =  e(v.,  —  Vi)  , 

where  e  is  a  constant,  less  than  unity,  which  is  called  the  Coefficient 
of  Restitution.  This  condition  asserts  that  the  relative  speed  of 
separation  of  the  two  bodies  is  less  than,  but  is  proportional  to, 
their  relative  speed  of  approach.  It  ceases  to  be  true  if  the  distor- 
tion produced  by  the  impact  is  too  great. 

If  the  bodies,  after  impact,  move  together  with  a  common  speed  V, 
the  first  of  these  equations  becomes 


This  principle  is  employed  in  the  Ballistic  Pendulum,  which  is 
used  to  determine  the  speed  of  a  cannon  ball  or  of  a  rifle  bullet.  In 
this  case  the  mass  of  the  pendulum,  m.2,  is  very  large  in  comparison 
with  the  mass  m1  of  the  bullet,  and  v.2  is  zero.  The  large  relative 
value  of  m.2  ensures  that  the  two  masses  are  moving  together  with 
the  common  speed  V  before  the  pendulum  has  been  sensibly  de- 
flected from  the  vertical.  The  value  of  V  is  found  by  observing  the 
distance  through  which  the  pendulum  swings.  [From  this,  the 
height  h  through  which  the  centre  of  inertia  (§  69)  is  raised  is 
obtained,  and  then  (§  42)  we  get  V  =  */%gliJ\ 

69.  Centre  of  Inertia.  —  In  a  material  system  composed  of  masses 
ma,  w0,  etc.,  we  can  always  find  a  point  such  that  the  product  of  its 
distance  from  any  plane  into  the  sum  of  the  separate  masses  is  equal 
to  the  sum  of  the  products  of  each  separate  mass  into  its  own  dis- 
tance from  that  plane. 

Let  2(m)  denote  the  sum  of  the  masses,  and  let  2(md)  denote  the 
sum  of  the  products  of  each  mass  into  its  distance  d  from  one  given 
plane.  We  then  can  obviously  find  the  distance  D  from  this  plane 
such  that 

S(wd),  .......  (a), 


MATTER    IN    MOTION.  89 

for  this  is  a  single  equation  in  one  unknown  quantity  D.  Let  this 
be  done  for  other  two  planes,  neither  pair  of  the  three  being 
parallel,  and  we  get  a  fixed  point,  which  satisfies  the  condition  for 
these  three  planes. 

Let  us  suppose,  for  convenience,  that  the  three  planes  are  at  right 
angles  to  each  other,  and  that  the  lines  of  intersection  are  taken  as 
the  axes  of  x,  y,  and  z  respectively.  We  have  then  the  three 
equations  similar  to  the  above, 


Now,  multiplying  these  equations  respectively  by  any  quantities 
X,  ft,  i>,  and  adding,  we  get 


But  X,  //,  and  v  may  be  the  direction-cosines  of  the  normal  to  any 
plane  passing  through  the  intersection  of  the  three  given  planes,  in 
which  case  the  quantity  \x+\iy+vz  is  the  perpendicular  upon  this 
plane  from  any  point  whose  co-ordinates  are  x,  ?/,  z.  Hence 
equation  (a)  is  true  for  any  plane  which  passes  through  the  intersec- 
tion of  the  three  given  planes. 

But  equation  (a)  is  still  satisfied  if  we  increase  D  and  d  by  any 
constant  quantity  h,  for  this  simply  adds  on  2(m)h  to  each  side  ; 
that  is  to  say,  (a)  holds  for  any  plane  parallel  to  a  given  one  for 
which  it  is  true.  Hence,  it  holds  for  all  planes. 

The  point  so  found  is  called  the  Centre  of  Inertia  of  the  given 
set  of  material  particles. 

By  taking  the  time-rate  of  variation  of  the  quantities  in  the  above 
equation  we  obtain 

2(m)D  =  2(w<Z) 
and 


The  former  tells  us  that  the  momentum  of  the  system  in  any  given 
direction  is  equal  to  the  momentum,  in  that  direction,  of  a  single 
mass,  equal  to  the  sum  of  the  separate  masses,  moving  so  as 
always  to  be  situated  at  the  centre  of  inertia.  The  latter  asserts 
that  the  change  of  motion  of  the  centre  of  inertia  of  any  set  of 
disconnected  particles  produced  by  the  action  of  separate  forces 
on  the  separate  masses  is  the  same  as  if  these  forces  had  been 
applied  to  a  mass,  equal  to  the  total  mass,  condensed  at  the  centre  of 
inertia. 

In  consequence  of  the  equality  of  action  and  reaction  between 


90 


A    MANUAL    OF    PHYSICS. 


material  particles,  we  see  that  the  motion  of  the  centre  of  inertia 
of  any  connected  set  of  particles  is  not  affected  by  their  mutual 
action ;  and  that,  in  the  case  of  a  rigid  body,  we  may  suppose  the 
whole  mass  to  be  condensed  at  the  centre  of  inertia,  and  to  be  acted 
upon  by  the  resultant  force.  In  other  words,  the  equations  of 
translational  motion  and  equilibrium  of  a  rigid  body  may  be  made 
identical  with  those  already  given  in  §  65  for  a  material  particle. 

70.  Moment  of  a  Force  and  of  Inertia. — The  Moment  of  a  Force 
as  regards  rotation  about  an  axis  perpendicular  to  its  direction  is  the 
product  of  the  force  into  the  shortest  distance  between  its  line  of 
action  and  the  axis. 

A  pair  of  parallel,  equal,  and  oppositely  directed,  forces  is  called  a 
couple.  The  moment  of  a  couple  about  any  axis  perpendicular  to 
the  plane  in  which  the  forces  act  is  equal  to  the  product  of  either 
force  into  the  perpendicular  distance  between  the  lines  of  action  of 
the  two.  Let  r  be  this  distance,  and  let  F  be  the  common  value  of 
the  forces,  while  P  is  the  intersection  of  any  perpendicular  axis  with 


I   F 


F'' 


FIG.  49. 


the  plane  in  which  the  forces  act.    If  x  is  the  perpendicular  distance 
from  P  to  the  line  of  action  of  one  force,  r-x  is  the  perpendicular 
distance  from  P  to  the  line  of  action  of  the  other.     Hence,  the  sum  of 
the  moments  of  the  forces,  which  is  the  moment  of  the  couple,  is 
Fx+F  (r-x)  =  Fr. 

Now  we  may  write  Fr  =  mar,  where  m  is' the  mass  acted  upon, 
and  a  is  the  linear  acceleration.  But  a  =  a,r,  where  w  is  the  angular 
acceleration.  Hence 

Fr=wra«. 

Three  independent  equations  of  this  type  completely  specify  the 
rotational  motion  of  the  given  mass. 

The  quantity  mr*  is  called  the  Moment  of  Inertia  of  the  mass  m 
about  the  given  axis.  Multiplying  each  side  of  the  equation  by  M<lt 
and  forming  the  integrals,  we  get 


MATTER    IN    MOTION.  91 

/ 

if  dQ/dt=io,  so  that  9  is  the  whole  angle  through  which  the  mass 
has  turned.  But  since  rw  =  v,  the  linear  speed,  the  quantity  on  the 
right-hand  side  of  the  equation  is  the  kinetic  energy  of  rotation. 
Hence,  we  see  that  the  kinetic  energy  of  rotation  acquired  under  the 
action  of  a  given  couple  is  proportional  to  the  angle  through  which 
the  mass  has  turned. 

If  we  are  not  dealing  with  a  single  particle,  we  must  write  the 
sum  2(rar2)  instead  of  mr2  in  the  above  equation.  But  we  may  still 
put  the  equation  in  the  same  form  as  before  by  writing 


which  is  clearly  allowable,  since  k  (called  the  radius  of  gyration)  is 
the  only  unknown  quantity. 

As  an  example,  we  shall  investigate  the  motion  of  a  cylinder 
rolling  (not  sliding)  down  a  plane  inclined  at  an  angle  a  to  the 
horizon.  Let  r  be  the  radius  of  the  cylinder,  and  let  Jc  be  its  radius 
of  gyration.  The  distance  through  which  the  cylinder  descends 
when  it  turns  through  an  angle  0,  is  s  =  rB  sin  a.  If  m  is  the  mass 


FIG.  50. 

of  the  cylinder,  the  work  done  by  gravity  is  mgs=mgr9  sin  «.  This 
must  be  equal  to  the  gain  of  kinetic  energy.  The  energy  in  the 
rotational  form  is  ±mW02,  and  that  in  the  translational  form  is 
^m(rO)'2.  Hence,  v(  =  rti)  being  the  speed  of  motion  down  the 
P^ne,  . 


Had  there  been  no  rotation,  we  should  have  had  v2  =  1gs;  but  the 
speed  of  linear  motion  has  been  decreased  because  the  potential 
energy  became  transformed  in  part  into  energy  of  rotation. 

71.  Further  Discussion  of  Moment  of  Inertia.  —  The  moment  of 
inertia  of  a  body  about  any  axis  is  equal  to  its  moment  of  inertia 
about  a  parallel  axis  through  the  centre  of  inertia,  together  with  the 
moment  of  inertia,  about  the  original  axis,  of  a  mass,  equal  to  the 
whole  mass,  condensed  at  the  centre  of  inertia. 

The  moment  of  inertia  is 


92 


A    MANUAL    OF    PHYSICS. 


Transfer  the  origin  to  P  (Fig.  51),  the  centre  of  inertia,  the  co- 
ordinates of  which  are  «,  /5.  Let  £,  rj  be  the  co-ordinates  of  any 
point  referred  to  parallel  axes  through  P.  Then 


since  2(w£),  2(m»/)  vanish  by  the  properties  of  the  centre  of  inertia. 
To  illustrate  the  importance  of  this  result  we  shall  proceed  to  find 
the  moment  of  inertia  of  a  cylindrical  rod,  of  length  2/  and  radius 
a,  about  an  axis  drawn  through  its  centre  perpendicular  to  its  length. 
Consider  a  circular  disc  of  the  rod  of  infinitesimally  small  thickness 
dli.  The  moment  of  inertia  of  this  disc  about  the  axis  of  the  rod  is 


FIG.  51. 

2(mr2),  where  r  is  the  distance  of  the  elementary  mass  m  from  the 
axis,  and  the  summation  extends  from  r  =  o  to  r—a.  If  p  is  the 
density  of  the  rod,  we  may  write  %*.rdrpdh  instead  of  m  ;  for  %vrdr 
is  the  area  of  a  small  circular  ring  of  the  disc,  so  that  litprdrdli 
is  the  mass  of  a  small  annular  portion  of  the  disc.  The  moment 
of  inertia  of  this  part  is  therefore  27rpd7ir3dr,  and  the  integral  of 
this  from  r  =  o  to  r  =  a  is  the  moment  of  the  whole  disc.  It  is 
therefore  ^irpdlia*. 

Now,  since  the  moment  of  the  whole  disc  about  a  central  perpen- 
dicular axis  is 


where  x  and  y  are  the  co-ordinates  of  m  referred  to  any  two  mutually 
rectangular  central  axes  in  the  plane  of  the  disc,  and  since  2(w«r-) 
and  2(m7/2)  are  respectively  the  moments  of  inertia  of  the  disc 
about  the  axes  of  x  and  y,  we  see  that  the  moment  of  the  disc 
(or  of  any  plane  figure)  about  an  axis  in  its  own  plane,  drawn 
through  its  centre  of  inertia,  is  one-half  of  its  moment  about  a 
perpendicular  axis  through  its  centre  of  inertia. 

The  moment  of  the  disc  about  a  central  axis  in  its  plane  is  there- 
fore ^irpdha4.     And,  if  h  be  the  distance  of  the  centre  of  the  disc 


MATTER   IN    MOTION. 


93 


from  the  centre  of  the  rod,  the  moment  of  inertia  of  the  whole 
mass  (irpa-dh)  of  the  disc,  supposed  to  be  condensed  at  its  centre, 
about  a  line  passing  through  the  centre  of  the  rod  and  perpendicular 
to  its  length,  is  7rpa'2dhlr.  Hence  the  moment  of  inertia  of  the 

disc  about  this  line  is  7rp<xM-  +W\dli.  And,  if  we  sum  the  moments 

of  all  such  discs  from  li  =  o  to  h  =  I,  we  get  half  the  moment  of 
inertia  of  the  whole  rod.  Taking  twice  the  integral  of  this  quan- 
tity between  these  limits,  we  find  that  the  required  moment  is 

Iwola1  (f +  1);  that  is,  M(  5+^"),  where  M  is  the  whole  mass  of 
\4       o  /  \o       4  / 

the  rod. 

72.  Rotational  Equilibrium. — There  is  no  rotation  about  a  given 
axis  when  2(Fr),  taken  with  reference  to  that  axis,  is  zero.  Hence, 
the  condition  for  rotational  equilibrium  is  that  the  sum  of  the 
moments  of  all  the  forces  about  three  non-parallel  axes  shall  vanish. 
The  two  following  examples  will  serve  to  illustrate  this  point. 

(1)  A  uniform  ladder  (Fig.  52),  of  length  2Z,  rests,  in  a  vertical 
plane,  upon  the  ground,  and  a  vertical  wall.  Find  the  limiting  posi- 


tion  of  equilibrium,  the  co-efficients  of  friction  becween  the  ladder 
and  the  ground,  and  between  the  ladder  and  the  wall,  being  p  and  n' 
respectively. 

Let  «  be  the  inclination  of  the  ladder  to  the  ground.  Under  the 
given  conditions,  there  is  no  possibility  of  motion  except  in  the 
given  vertical  plane.  Hence,  there  are  only  three  degrees  of 
freedom,  viz.,  two  degrees  as  regards  translation,  and  one  as  regards 
rotation.  The  ladder  will  be  in  equilibrium,  so  far  as  translation  is 
concerned,  if  the  sums  of  the  forces  acting  upon  it  in  any  two 
mutually  perpendicular  directions  are  zero.  But  it  is  convenient  to 


94  A    MANUAL    OF    PHYSICS.  t 

choose  those  two  directions  which  will  lead  to  the  simplest  equations. 
We  might  choose  the  directions  along  and  perpendicular  to  the 
length  of  the  ladder,  but  each  of  the  consequent  equations  would 
involve  all  trie  five  forces  which  are  acting.  If  we  choose  the 
horizontal  and  vertical  directions,  the  equations  involve  respectively 
two  and  three  forces  only.  Therefore,  choosing  the  latter  directions, 
we  get 


where  S  and  R  are  the  normal  pressures  on  the  wall  and  the  ground 
respectively. 

The  third  relation  between  the  quantities  is  obtained  by 
equating  to  zero  the  sum  of  the  moments  of  the  various  forces 
about  any  axis  perpendicular  to  the  plane  of  motion.  The  simplest 
equation  is  obtained  by  choosing  the  axis  passing  through  a  point 
(either  end  of  the  ladder),  which  lies  on  the  lines  of  action  of  the 
greatest  possible  number  of  forces  ;  the  reason  being  that  the 
moments  of  these  forces  are  then  zero.  Taking  the  lower  end 
we  get 

mgl  cos  «  =  (S"sin  a+//S  cos  a)2Z, 
that  is 

mg  cos  rt  =  2S(sin  «+/*'  cos  a), 

which  is  independent  of  the  length  of  the  ladder. 

If  we  suppose  the  weight  of  the  ladder,  and  the  values  of  /<  and  /*' 
to  be  given,  we  may  eliminate,  by  means  of  these  three  equations, 
the  quantities  S  and  R,  and  so  obtain  an  equation  giving  a  in  terms 
of  known  quantities. 

(2)  A  pendulum,  of  length  I  and  mass  m,  rotates  about  a  vertical 
axis  with  constant  angular  velocity  w.  Express  w  in  terms  of  </, 
the  value  of  gravity,  and  of  7t,  the  height  of  the  cone  which  the 
pendulum  describes. 

Let  9  be  the  angle  which  the  pendulum  makes  with  the  vertical. 
The  '  centrifugal  force  '  acting  perpendicular  to  the  axis  is  muPl  sin  0. 
The  portion  of  this  which  is  perpendicular  to  the  string  of  the 
pendulum,  and  which  acts  so  as  to  prevent  decrease  of  0,  is  un.rl 
sin  9  cos  9.  The  part  of  the  weight  which  acts  in  the  same  line,  but 
inwards  so  as  to  decrease  0,  is  mgsmQ.  Hence  the  condition  for 
equilibrium  is 

w3Z  cos  Q  =  M2li  =  g. 

This  gives  the  required  expression  for  w,  and  shows  that  the  angle  9 
is  constant. 

73.  Propagation  of  Motion  through  a  Non-Rigid  Solid.  —  As  a 


MATTER    IN    MOTION. 


95 


single  example  of  the  motion   of   a   non-rigid  solid  we  shall  now 

discuss  the  problem  of  the  passage  of  a  wave  along  a  stretched  cord. 

Let  us  suppose  the  cord  to  be  enclosed  in  a  smooth  hollow  tube, 

and  to  be  drawn  through  it  in  the  direction  of  the  arrow  with  speed  v. 


FIG.  53. 

The  tension  T  of  the  cord  will  be  uniform  throughout  since  the 
tube  is  smooth. .  The  pressure  which  the  cord,  if  not  in  motion, 
would  exert  upon  a  part  of  the  tube  where  the  radius  of  curvature 
is  r  would  be  T/r.  Let  the  cord  be  in  contact  throughout  a  circular 
arc  PQ  which  subtends  an  angle  0  at  the  centre  O.  Then,  if  OE 
bisects  0,  the  resolved  part  of  the  tensions  along  EO  is  2T  sin  0/2. 
If  0  is  small  this  becomes  T0  =  T  .  PQ/E.  But  the  total  pressure 
is^PQ,  where  p  is  the  pressure  per  unit  length  of  the  circle.  Hence 


R 


p  =  T/E  ;  that  is  to  say  p  is  proportional  conjointly  to  the  tension 
and  the  curvature. 

If  m  is  the  mass  per  unit  length  of  the  cord,  mv-fr  is  the  '  centri- 
fugal force  '  when  the  cord  is  in  motion  with  speed  v.  When  this 
is  equal  to  T/r,  i.e.,  when  T  =  mv2,  there  is  no  pressure  on  the 
surface.  The  value  of  v  which  satisfies  this  equation  is  totally 
independent  of  r,  the  radius  of  curvature.  Hence,  when  the  proper 
speed  is  reached  the  pressure  is  simultaneously  taken  off  all  parts 
of  the  smooth  tube  through  which  the  cord  runs  ;  and  the  tube, 
having  served  the  purpose  for  which  it  was  used,  might  now  be 
dispensed  with.  All  parts  of  the  cord  would  successively  take  the 
shape  which  the  tube  originally  impressed  upon  the  portion  within 
it.  And  also,  since  all  motion  is  relative,  if  the  cord  were  held 
fixed  with  the  given  constant  tension,  the  wave-form  would  run 


96  A    MANUAL    OF    PHYSICS. 

backwards  along  it  with  speed  v.  Hence  the  speed  with  which 
any  disturbance  will  run  along  a  cord  stretched  with  tension 
Tis 


when  m  is  the  mass  per  unit  length. 

Simple  as  the  above  proof  (due  to  Thomson  and  Tait)  is,  the 
following,  deduced  from  the  principle  of  dynamical  similarity,  is  at 
least  as  simple. 

The  radius  of  curvature  at  similar  parts  of  similar  waves  is  pro- 
portional to  the  length  I  of  the  wave.  The  pressure  per  unit  of 
length  is  therefore  proportional  to  T/Z,  so  that  the  pressure  per 
similar  length  is  proportional  to  T.  Also  the  mass  per  similar 
length  is  ml,  and  hence  we  get  the  dimensional  equation 


where  t  is  the  periodic  time  in  which  the  wave  length  I  is  described. 
But  IJ  t  is  the  speed  of  propagation,  which  is  therefore  equal  to 
A^T/A/m^  if  we  adopt  the  definition  of  force  given  in  ^  64. 

74.  Motion  of  a  Perfect  Fluid.  —  A  fluid  may  be  set  in  motion 
by  the  action  either  of  forces  which  act  throughout  its  volume  (for 
example,  gravitational  forces)  or  of  forces  which  are  applied  to  its 
surface  (such  as  external  pressure). 

A  perfect  fluid  is  defined  as  a  fluid  in  which  the  pressure  is 
always  perpendicular  to  the  surfaces  of  contact.  It  may  other- 
wise be  defined  as  a  fluid  which  is  entirely  devoid  of  internal 
friction.  Such  a  fluid  does  not  exist  in  nature,  but  we  may  deduce 
various  results  regarding  the  motion  of  perfect  fluids  which  will  be 
very  nearly  true  for  actual  fluids  which  are  moving  with  sufficient 
slowness.  [When  any  fluid,  whether  perfect  or  not,  is  at  rest,  the 
pressure  is  always  perpendicular  to  the  surfaces  of  contact.] 

Consider  a  little  cube  with  edges  dx,  dy,  dz,  parallel  to  the  axes 
in  a  fluid  of  density  p.  The  mass  of  this  little  cube  is  pdjcdydz, 
and  its  acceleration  of  momentum  parallel  to  the  «c-axis  is  pxdxdydz  ; 
and  this  is  equal  to  the  sum  of  the  forces  acting  upon  the  little  mass 
in  that  direction.  Let  X  be  the  force  per  unit  of  mass  which  is 
acting  upon  it,  and  let  p  be  the  pressure  per  unit  of  area.  The  total 
pressure  on  the  face  of  the  cube  next  the  origin  is  jpdydz,  and  this 
acts  outwards  along  the  #-axis.  When  x  changes  by  the  amount 
dx,  p  will  alter  to  p  +  dp,  so  that  the  pressure  acting  inwards  along 


MATTER   IN    MOTION.  97 

the  axis   is    (p  -\-  dp)dydz.      Hence   the  total  outward  pressure  is 

dt) 

—  dpdydz,  or  as  we  may  write  it,  -  -j-dxdydz.     Hence  we  have  as 

dx 

the  equation  of  motion  parallel  to  the  ,r-axis 

dp 


Similarly  ..  d 

W=PY-^, 

and  jn 


[It  must  be  carefully  noticed  that  x,  y,  and  #,  represent  the  total 
component  accelerations,  which  may  vary  independently  with  the 
time  and  with  the  position  of  the  small  mass  :  in  short,  we  are 
supposed  to  follow  the  given  portion  of  the  fluid  in  its  motion.] 

As  an  example  we  shall  investigate  the  motion,  under  gravity,  of 
a  fluid  which  escapes"  through  a  small  orifice  in  the  side  of  a 
vessel,  the  depth  of  the  opening  below  the  free  surface  of  the  liquid 
being  z.  Take  the  origin  at  the  free  surface,  the  axis  of  z  being 
drawn  downwards,  and  the  axes  of  x  and  y  being  horizontal.  The 
equations  of  motion  are 

dp       "          dp      "  dp 

P»--^I*=^:  <"-(*-£' 

g    being    the    acceleration    due    to    gravity.       Multiplying    these 
equations  by  xdt,  ydt,  zdt,  respectively,  and  adding  we  get, 


p(xdx  +  ydy  +  zdz)  =  pgdz 
The  integral  of  this  is 

where  v  =  v^+  7)2  +^2  ™  tne  speed  of  motion  of  the  fluid  and 
pu  is  the  pressure  on  the  free  surface  of  the  liquid  since  it  is  the 
value  of  p  when  z  =  o  and  v  =  o,  which  is  practically  the  case  when 
the  area  of  the  opening  by  which  the  fluid  escapes  is  very  small  in 
comparison  with  the  free  surface  of  the  liquid. 

The  pressure  just  outside  the  opening  is  pot  and  hence  we  have 


98  A   MANUAL    OF    PHYSICS. 

v-  =  tyz.  That  is,  the  speed  is  that  which  would  be  acquired  in 
a  free  fall  from  rest  under  gravity  through  a  distance  equal  to  the 
depth  of  the  opening  below  the  surface  of  the  liquid. 

This  result  may  readily  be  deduced  by  considerations  regarding 
the  energy  of  the  liquid.  The  kinetic  energy  of  a  quantity  m  of  the 
the  escaping  liquid  is  |wy2.  But  this  energy,  which  the  escaping 
liquid  carries  away  with  it,  is  at  once  restored  if  we  simply  pour  the 
liquid  back  again  into  the  vessel.  And  the  work  done  in  raising  the 
liquid  through  the  height  %  is  mgz.  Hence,  by  the  principle  of 
conservation  of  energy,  we  have,  as  before,  v2  =  tyz. 

Equation  (1)  shows  that,  in  a  moving  fluid,  the  pressure  is  least 
where  the  speed  is  greatest.  Hence  there  is  less  pressure  in  the 
interior  of  a  moving  jet  of  fluid  than  there  is  at  the  outside.  Thus 
objects  immersed  in  the  fluid  will  be  pressed  inwards  to  the  centre 
of  the  jet.  This  explains  the  support  of  a  light  body  in  a  vertical 
jet  of  water  or  of  air. 

75.  Equilibrium  of  a  Fluid.  —  From  the  equations  of  motion  of 
a  fluid  we  at  once  get,  as  a  special  case,  the  conditions  of  equilibrium. 
These  are 

dp,  dp  dp 

p  A.  =  =—  ;  p  i  =  -y-  ;  $L  =  —  -  . 

dx  dy  dz 

If  no  external  forces  act  upon  the  liquid,  we  have  the  equations 

dp  dp  dp 

j  r  =°;  :r-  =  °;  :r  =  0> 
dx  dy  Az 

which  simply  assert  that  the  pressure  has  a  constant  value  at  all 
points  of  the  liquid. 

If  we  assume  that  the  origin  is  at  the  surface  of  the  liquid,  that 
gravity  acts,  and  that  the  axis  of  z  is  drawn  downwards,  the  equa- 
tions are 

dn  dp          dp 

~i  <-  =  °  >  -T  =  °  >    i=P9' 
dx  dy  dz 

The  first  two  assert  that  the  pressure  is  constant  throughout  a 
horizontal  plane,  and  the  last  shows  that  it  increases  uniformly 
with  the  depth.  The  integral  is 


This  might  have  been  obtained  from  (1)  of  last  section  by  making 
v  =  o» 

76.  Propagation  of  Surf  ace-Waves  in  Liquids.  —  We  shall  assume, 


MATTER    IN    MOTION.  99 

for  the  sake  of  simplicity,  that  the  waves  are  all  similar,  and  that 
their  ridges  are  parallel  equi-distant  straight  lines.  The  principle  of 
dynamical  similarity  then  enables  us  to  deduce  easily  the  law  of 
propagation. 

When  the  waves  are  propagated  by  gravity,  the  forces  are  propor- 
tional to  the  density  of  the  liquid,  to  the  value  of  gravity,  and  to 
the  square  of  the  wave-length  ;  and  the  masses  are  proportional  to 
the  density,  and  to  the  square  of  the  wave-length.  Hence  the 
dimensional  equation 

f=ml/t2, 

(§67),  becomes  7o       aV  .  I 


or  g=ll&, 

which  gives  v2=gl. 

In  these  equations  p  represents  the  density  of  the  liquid,  and  the 
other  symbols  have  the  usual  significations.  We  see,  therefore,  that 
the  speed  of  propagation  is  proportional  conjointly  to  the  square 
roots  of  the  wave-length,  and  of  the  acceleration  due  to  gravity. 
Such  waves  are  called  oscillatory  or  free  waves. 

In  the  above  case,  it  is  assumed  that  the  depth  of  the  liquid  is 
very  large  in  comparison  with  the  length  of  the  waves.  When  the 
depth  is  very  small  in  comparison  with  the  wave-length,  the  above 
equations  still  apply,  provided  that  I  represents  the  depth  of  the 
liquid.  For,  when  similar  waves  are  propagated  in  liquids  of 
different  depths  (the  similarity  having  reference  to  the  depth),  we 
see  that  similar  masses  are  proportional  to  the  squares  of  the  depths, 
while  the  ranges  of  vertical  motion  are  proportional  directly  to  the 
depth.  Hence  the  speed  of  propagation  of  such  waves,  which  are 
called  long  or  solitary  waves,  is  proportional  conjointly  to  the  square 
roots  of  the  depth  and  of  the  acceleration  due  to  gravity. 

In  the  propagation  of  ripples,  surface-tension  (Chap.  X.)  is  much 
more  effective  than  gravitation  is.  If  T  represents  the  surface- 
tension,  while  I  represents  the  wave-length,  the  pressure  per  unit 
area  of  the  surface  is  proportional  to  T/Z.  Hence  the  pressure  per 
similar  area  is  proportional  to  T,  for  we  are  not  concerned  with 
lengths  measured  parallel  to  the  ridges  of  the  waves.  The  similar 
masses  are  proportional  to  p  and  .to  Z",  and  so  we  get 


yu 

The  speed  of   propagation  of    a  ripple  is  therefore  proportional 

7 9 


100  A    MANUAL    OF    PHYSICS. 

directly  to  the  square  root  of  the  surface-tension,  and  inversely  to 
the  square  root  of  the  product  of  the  density  into  the  wave-length. 
Thus  we  see  that  ripples  run  faster  the  smaller  they  are,  while 
oscillatory  waves  run  faster  the  larger  they  are.  Hence  there  is  a 
certain  size  of  wave  (about  two-thirds  of  an  inch  in  length  in  the 
case  of  water)  which  runs  slowest.  Smaller  waves  run  more 
quickly  because  the  effect  of  surface-tension  preponderates ;  larger 
waves  run  more  quickly  because  of  the  increased  effect  of  gravity. 


CHAPTEE  VII. 

PROPERTIES    OF    MATTER. 

77.  Definitions  of  Matter. — We  are  now  in  a  position  to  give  one 
or  two  provisional  definitions  of  matter — provisional,  because  we 
cannot  yet  say,  possibly  may  never  be  able  to  say,  what  matter 
really  is.  It  may  be  defined  in  terms  of  any  of  its  distinctive 
characteristics.  We  may  say  that  Matter  is  that  which  possesses 
Inertia.  Or  again,  since  we  have  no  knowledge  of  energy  except 
in  association  with  matter,  we  may  assert  that  Matter  is  the  Vehicle 
of  Energy.  Another  statement  (which,  from  the  results  of  Chap.  I. 
and  §  62,  we  see  to  be  an  objectionable  form  of  the  latter)  is 
that  '  matter  is  that  which  exerts,  or  can  be  acted  upon,  by  force.' 
Further  knowledge  would  probably  make  it  evident  that  these  three 
definitions  are  merely  differently  worded  statements  of  the  same 
fact. 

78.  States  of  Matter. — Matter  is  usually  spoken  of  as  existing 
in  three  different  states — the  solid,  the  liquid,  and  the  gaseous. 

A  portion  of  matter  in  the  solid  state  possesses  a  definite  form  of 
its  own,  and  considerable  force  has  to  be  applied  in  order  to  produce 
an  appreciable  change  in  the  form.  When  in  the  liquid  state, 
matter,  on  the  other  hand,  possesses  no  definite  form  of  its  own, 
and  can  be  made  by  application  of  the  slightest  force  to  change 
whatever  form  it  happens  to  have.  A  similar  statement  holds  in 
the  case  of  a  gas  or  vapour.  But  a  gaseous  body  differs  from  a 
liquid  in  that  its  volume  is  limited  only  by  the  volume  of  the  closed 
vessel  in  which  it  is  contained ;  while  the  volume  of  a  given  quan- 
tity of  liquid  is  perfectly  definite,  under  given  physical  conditions, 
however  large  the  containing  vessel  may  be. 

Still — although  these  rules  are  of  general  applicability — it  must 
not  be  supposed  that  there  is  any  hard  and  fast  distinction  between  a 
solid  and  a  liquid,  or  between  a  liquid  and  a  vapour  :  possibly  (but  there 
is  no  experimental  proof  of  the  truth  of  this  statement)  there  maj"  be, 


102  A   MANUAL   OF   PHYSICS. 

under  certain  physical  conditions,  no  sharp  line  of  demarcation 
betwet-ji  the,  solid  and.the  vaporous  states  of  matter. 

As  the  temperature  of  sealing-wax  is  gradually  raised,  the  sub- 
stance slowly  passes  from  the  solid  into  the  liquid  condition,  and,  for 
some  time,  we  cannot  strictly  call  it  either  a  liquid  or  a  solid.  The 
transition  from  ice  to  water  seems  to  occur  suddenly,  but  analogy 
would  indicate  that  the  process  is  really  a  continuous  one.  We  shall 
also  find  (Chap.  XXIII.)  that  the  passage  from  the  vaporous  to  the 
liquid  condition  may  be  made  without  break  of  continuity. 

[A  very  notable  and  extremely  important  example  of  the  non- 
rigidity  of  the  above  distinctions  occurs  in  the  case  of  shoemakers' 
wax.  This  substance  so  far  resembles  a  brittle  solid  that  it  will 
break  into  splinters  under  the  blow  of  a  hammer ;  and  yet,  under 
the  action  of  slight  long-continued  forces,  it  can  be  moulded  into  any 
shape  we  please.] 

The  extreme  form  of  the  gaseous  condition,  which  is  known  as  the 
'  ultra-gaseous '  or  '  radiant '  state  of  matter,  will  be  discussed  in 
Chap.  XIII. 

79.  General  Properties. — Certain  properties  are  common  to  all 
portions  of  matter  in  whatever  state  or  physical  condition  they  may 
be,  and  are,  therefore,  called  general  properties. 

Chief  among  them  is  the  already-mentioned  property  of  inertia, 
which  requires  no  further  discussion  at  present. 

Again,  all  matter  occupies  space ;  and  we  consequently  say  that 
it  has  the  property  of  extension.  This  subject  has  been  considered 
in  Chap.  III.  But  the  occupancy  of  space  further  involves  the  idea 
of  form,  and  so  we  recognise  form  as  a  property  of  matter.  There 
is  little  more  to  be  said  on  this  point  except  in  the  case  of  the  form 
of  crystallised  bodies,  to  the  consideration  of  which  a  considerable 
part  of  Chap.  XII.  will  be  devoted. 

So  far  as  we  know  any  one  portion  of  matter  occupies  a  given  por- 
tion of  space  to  the  utter  exclusion  of  all  other  matter.  This  cer- 
tainly holds  in  the  case  of  any  visible  portion.  Hence  we  look  upon 
impenetrability  as  a  general  property  of  matter.  But  the  quality  of 
impenetrability  does  not  interfere  with  inter-penetration  of  matter, 
the  possibility  of  which  depends  upon  the  existence  of  pores  in  any 
finite  portion  of  matter.  The  corresponding  property  is  called 
porosity.  All  matter  is  more  or  less  sponge-like  in  structure,  the 
space  in  the  so-called  '  internal '  pores  being  in  reality  external  to 
the  material  of  the  body.  In  the  case  of  substances  such  as  cork, 
wood,  coarse  sandstone,  etc.,  the  porosity  is  very  evident.  The 
porosity  of  metals  is  shown  by  the  fact  that  gases  can  pass  through 
them.  Thus  palladium  has  a  remarkable  power  of  absorbing  or 


PROPERTIES   OF   MATTER,  103 

'  occluding '  hydrogen ;  carbonic  oxide  passes  readily  through  red-hot 
iron ;  and  gases  formed  by  the  decomposition  of  electrolytes  pass 
through  the  metallic  electrodes.  Bichromate  of  potassium  passes 
into  the  pores  of  glazed  earthenware,  and,  crystallizing  inside, 
gradually  breaks  up  the  substance.  The  porosity  of  liquids  is 
evidenced  by  their  absorption  of  gases. 

Vitreous  bodies  alone  have  not  yet  been  directly  shown  to  be 
porous;  but  we  may  fairly  conclude  that  we  have  not  yet  found 
the  proper  method  of  testing  the  point,  and  that,  the  proper  method 
being  found,  they  too  will  prove  to  be  no  exception  to  the  rule. 

A  very  noteworthy  example  of  interpenetration  occurs  in  the 
alloying  of  certain  metals,  such  as  tin  and  copper.  The  bulk  of 
this  alloy  is  considerably  less  than  the  sum  of  the  bulks  of  its 
constituents.  This  phenomenon  does  not  occur  when  gold  and 
silver  are  alloyed,  and  for  this  reason  only  was  Archimedes'  famous 
test  of  the  impurity  of  Hiero's  crown  conclusive.  A  certain  weight 
of  pure  gold  had  been  given  to  a  smith  for  the  purpose  of  making 
the  crown ;  but  it  was  suspected  that  he  had  abstracted  some  of 
the  gold,  replacing  it  by  an  equal  weight  of  silver.  Archimedes 
knew  that,  weight  for  weight,  silver  is  bulkier  than  gold;  and 
hence  he  concluded  that  the  crown  would  be  bulkier  than  the  given 
amount  of  pure  gold  if  silver  had  been  used  as  an  alloy.  The 
problem  which  he  required  to  solve  was  therefore  the  determination 
of  the  bulk  of  a  solid  which  was  so  irregular  in  shape  that  no 
method  of  estimation  by  direct  measurement  was  applicable.  He 
determined  this  by  measuring  the  volume  of  the  water  which 
the  crown  displaced.  Had  contraction  taken  place,  the  bulk  and 
weight  of  the  alloy  might  have  been  the  same  as  those  of  the  pure 
gold. 

Another  property  of  all  matter  is  divisibility.  The  question  of 
the  infinite  divisibility  of  matter  will  be  further  alluded  to  in  the 
chapter  on  the  constitution  of  matter.  In  the  meantime  we  are 
only  concerned  with  examples  of  extreme  division.  Many  such 
occur  readily.  Films  of  gold  and  of  other  metals  may  be  made  so 
thin  as  to  be  transparent.  A  film  of  gold  precipitated  by  chemical 
means  and  burnished  so  that  it  forms  a  continuous  sheet  may  be  of 
no  greater  thickness  than  one  ten-millionth  part  of  an  inch.  A 
quartz -fibre  may  be  made  so  fine  as  to  be  utterly  invisible.  The 
vapour  from  a  particle  of  sodium  will  tinge  a  flame  continuously  of 
a  deep  orange  colour  for  hours  at  a  time.  A  single  drop  of  a  strongly 
coloured  liquid  will  continuously  tinge  a  very  large  quantity  of 
water,  and  its  presence  may  be  made  evident  by  chemical  means 
long  after  the  eye  ceases  to  detect  it.  Further  examples  of  the 


104  A   MANUAL    OF    PHYSICS. 

extreme  smallness  of  portions  of  matter  will  be  given  in  the  chapter 
on  the  constitution  of  matter. 

All  matter  is  capable  of  having  its  volume  diminished  under 
pressure  to  a  greater  or  less  extent,  and  so  we  speak  of  its  compres- 
sibility. This  subject  will  receive  detailed  treatment  in  Chaps.  IX., 
X.,  and  XI. 

Again,  all  matter  is  deformable.  But  it  is  often  more  convenient 
to  speak  of  its  rigidity  than  of  its  deformability,  that  is,  the  pro- 
perty in  virtue  of  which  it  resists  deformation.  This  question,  too, 
will  be  discussed  subsequently. 

We  shall  also  afterwards  consider  more  specially  elasticity,  which, 
in  one  or  other  of  its  two  forms,  exists  in  all  kinds  of  matter ;  and 
viscosity,  that  is,  the  property  in  virtue  of  which  there  is  resistance 
to  relative  motion  of  the  particles  of  a  body.  Expansibility  will 
be  dealt  with  in  Chap.  XXII. 

Weight — which,  though  a  universal  property  of  matter,  may  be 
looked  upon  as  a  purely  accidental  property,  seeing  that  it  requires 
the  existence  of  two  separate  masses  in  order  that  it  may  appear — 
will  be  fully  considered  in  the  chapter  on  '  Gravitation.'  (See  also 
Chap.  VI.,  §  64.) 

80.  Special  Properties. — Many  other  properties  might  be  enume- 
rated which  are  conspicuously  present  in  some  substances,  and  are 
as  conspicuously  absent  from  others,  such  as  plasticity,  ductility, 
brittleness,  tenacity,  etc.     Again,  many  properties  refer  to  matter 
in  connection  with  special  forms  of  energy ;  for  example,  dispersive 
power,  thermal  and  electric  conductivity,  magnetic  permeability, 
translucency,  opacity,  etc.     Indeed,  all  the  properties  of  matter 
might  be  naturally  investigated  in  a  treatise  on  energy ;  for  we  have 
no  notion  of  what  might  be  the  properties  of  matter  devoid  of 
energy.     Most  probably  it  would  not  be  matter  at  all  in  the  sense 
in  which  we  use  the  word. 

81.  Specific  Properties. — Many  of  the  properties  of  a  body  depend 
upon  the  size  of  that  body.     Thus  a  portion  of  a  given  substance  is 
more  massive  than  another  portion  of  the  same  substance  in  pro- 
portion as  its  bulk  is  greater  than  the  bulk  of  that  other. 

It  is  frequently  very  essential  to  define  a  property  of  a  body  in 
such  a  way  as  to  make  it  independent  of  the  size  of  any  particular 
specimen.  For  example,  the  density  (specific  mass)  of  a  substance 
is  the  mass  per  unit  volume  of  that  substance ;  the  specific  gravity 
is  the  weight  of  unit  volume,  expressed  in  terms  of  that  of  water  as 
the  standard ;  the  specific  weight  might  be  defined  as  the  weight 
per  unit  volume,  expressed  in  absolute  units.  We  also  define 
rigidity,  viscosity,  etc.  (§§  108,  128)  as  specific  properties. 


PROPERTIES    OF    MATTER.  105 

82.  Molecules  and*  Atoms. — A  visible  portion  of  any  chemically 
compound  substance,  e.g.,  water,  may  be  divided  into  smaller  parts 
each  of  which  has  the  same  chemical  peculiarities  as  the  whole  had. 
These  parts  may  even  be  so  small  as  to  be  invisible  to  the  most 
powerful  microscope.     But  at  last  a  stage  is  reached  where  further 
division  cannot  occur  without  the  production  of  substances  which 
are    chemically   different   from   the   original  one.     These  smallest 
parts,  which  are  chemically  similar  to  the  whole,  are  called  mole- 
cules. 

Every  molecule  can  be  divided  further  into  what  are  termed  its 
constituent  atoms.  These  atoms  are  dissimilar  when  the  molecule 
is  complex  like  that  of  water :  they  are  all  precisely  similar  in  a 
simple  molecule,  such  as  that  of  hydrogen.  An  atom  is  an 
indivisible  part — that  is,  indivisible  by  any  means  at  present  at  our 
disposal.  In  all  probability,  it  is  really  compound. 

A  substance  wrhose  molecules  are  composed  entirely  of  similar 
atoms  is  called  an  '  elementary '  substance,  and  the  kind  of  matter 
composing  it  is  called  a  '  chemical  element.'  We  know  of  the 
existence  in  nature  of  only  a  comparatively  small  number  of 
elements. 

One  fact  of  the  utmost  importance  is  this — that  wherever  in  space 
it  may  be  situated,  or  howsoever  it  may  be  circumstanced  physi- 
cally, an  elementary  molecule  or  atom  has  absolutely  unalterable 
properties.  A  molecule  of  hydrogen  (for  example)  in  the  most 
distant  nebula  is  precisely  similar  to  a  molecule  of  that  substance 
on  the  earth's  surface.  We  shall  return  to  this  point  later  on. 

83.  Molecular  Forces. — A  considerable,  frequently  a  very  great, 
amount  of  work  is  necessary  in  order  to  separate  the  molecules  of  a 
body.     When  separated,  we  are  accustomed  to  say  that  they  possess 
potential  energy  of  molecular  separation — the  increase  of  potential 
energy  being  equivalent  to  the  work  spent  in  producing  the  separa- 
tion.   If  e,  the  energy,  increases  by  the  amount  de  when  the  distance  s 
between  the  molecules  is  increased  by  the  quantity  ds,  the  work  spent 
is  represented  also  by  the  quantity 

de=—ds. 
ds 

The  quantity  dejds — the  space-rate  of  variation  of  the  energy — 
is  called  the  molecular  force  against  which  the  work  is  done 
(see  §  29).  That  is  to  say,  we  figure  the  molecules  to  ourselves  as 
held  together  by  certain  forces  which  maintain  them  in  their 
relative  positions. 

The  work  done  in  separating  two  molecules  beyond  the  range  of 


106  A   MANUAL    OF   PHYSICS. 

their  mutual  forces  is  usually  great,  and  practically  the  whole  of  it  is 
performed  in  an  excessively  short  distance.  Hence  de/ds  is  very 
large  ;  and  so  it  is  said  that  the  molecular  forces  are  extremely  power- 
ful, but  that  they  are  insensible  at  sensible  distances.  In  confirmation 
of  this  we  may  firmly  press  together  two  leaden  bullets  at  a  part 
where  their  surfaces  have  been  freshly  cleaned.  They  will  cling 
together  so  that  the  lower  one  may  be  lifted  by  means  of  the  upper, 
the  magnitude  of  the  so-called  molecular  forces  at  the  surface  of 
contact  being  sufficient  to  overcome  the  gravitational  attraction  of 
the  whole  earth.  If  there  is  a  film  of  oxide  on  the  metal  the 
experiment  will  not  succeed,  the  thickness  of  the  film  being  too 
great  to  allow  of  appreciable  molecular  attraction.  (For  further 
statements  on  this  subject,  see  Chap.  XII.) 

Many  of  the  properties  of  matter  may  be  said  to  depend  on  the 
nature  of  the  molecular  forces.  Tenacity  depends  upon  the  extent 
to  which  these  forces  can  overcome  external  forces  which  tend  to 
draw  the  molecules  apart.  Malleability  is  a  property  essentially 
analogous  to  the  preceding.  It  is  the  property  in  virtue  of  which  a 
substance  may  be  extended  in  two  directions,  while  it  is  contracted 
in  a  direction  perpendicular  to  these  by  the  application  of  great 
pressure,  its  volume  remaining  practically  unaltered.  In  testing 
tenacity  the  extension  occurs  in  one  direction  only,  and  there  is 
contraction  in  all  directions  perpendicular  to  that  one.  The  brittle- 
ness  of  a  body  is  due  to  the  comparative  ease  with  which  the  mole- 
cules can  be  separated  beyond  the  range  at  which  their  mutual 
forces  are  appreciable.  Eigidity  depends  upon  the  ability  of  the 
molecular  forces  to  resist  alteration  of  the  relative  positions  of  the 
molecules  of  a  body.  We  therefore  recognise  two  kinds  of  rigidity 
— rigidity  as  regards  bulk,  and  rigidity  as  regards  form.  Viscosity 
depends  upon  the  ability  of  the  forces  to  resist  shearing  motion. 

84.  In  the  immediately  succeeding  chapters,  a  more  detailed 
examination  of  some  of  the  most  important  of  the  properties  alluded 
to  above  will  be  given.  Others  will  be  treated  as  occasion  may 
arise  subsequently. 


CHAPTEK  VIII. 

GRAVITATION. 

85.  ALL  bodies  in  the  neighbourhood  of  the  earth's  surface  possess 
potential  energy,  which,  when  circumstances  permit,  is  invariably 
changed  into  kinetic  energy  of  motion  towards  the  earth.  Hence 
we  say  that  each  body  is  '  attracted  '  to  the  earth  with  a  force  which 
is  termed  its  '  weight.' 

Bodies,  made  of  the  same  material,  may  be  roughly  judged  to 
be  heav3r  in  proportion  to  their  bulk,  i.e.,  in  proportion  to  the 
quantity  of  matter  contained  in  them.  But  we  need  not  rest 
content  with  a  roughly  approximate  rule,  for  the  law  that  weight 
is  proportional  to  mass  is  capable  of  as  rigid  proof  as  can  be  given 
by  the  most  accurate  physical  methods. 

In  the  first  place  every  body — except  in  so  far  as  the  resistance 
of  the  air  is  concerned — takes  the  same  time  to  fall  through  a  given 
distance.  In  other  words,  the  acceleration  of  motion  is  the  same 


in  all  cases.     Hence,  by  Newton's  Second  Law,  the  force  (weight) 
is  proportional  to  the  mass. 

The  fact  that  the  time  of  oscillation  of  a  simple  pendulum  is  inde- 
pendent of  the  mass  of  the  bob  furnishes  a  more  readily  obtainable 


108  A    MANUAL    OF    PHYSICS. 

proof  of  the  proportionality  of  weight  and  mass.  Let  9  be  the 
angle  through  which  the  pendulum  is  deflected  from  the  vertical. 
If  w  is  the  weight  of  the  body  the  force  acting  in  the  direction  of 
motion  is  w  sin  9  ;  which,  for  small  angles,  is  practically  wQ.  This 
must  be  equal  to  the  acceleration  of  momentum,  which  is  ml9, 
if  I  is  the  length  of  the  pendulum.  Hence  iuQ  =  mW\  and  there- 
fore (§  51) 


a  and  a  being  constants.     This  shows  that  the  motion  is  simple 
harmonic.     And  the  time  of  a  complete  oscillation  is 


2, 


v^ 


since  the  value  of  9  is  unaltered  if  we  increase  t  by  this  amount. 
Now  experiment  shows  that  this  quantity  is  constant  when  I  is 
constant.  Hence  w  is  proportional  to  m. 

Again,  a  body  has  the  same  weight  whether  its  surface  is  large 
or  small,  and  whether  it  is  in  a  single  lump  or  broken  up  into  parts. 
This  proves  that  the  outer  parts  do  not  screen  the  interior  parts 
from  the  action  of  gravitation.  Indeed,  perpetual  motion  could 
ensue  if  this  were  not  so  ;  for,  if  matter  were  placed  between  one 
half  of  a  vertically  mounted  wheel  and  the  earth,  the  other  half  of 
the  wheel  would  be  permanently  heavier. 

86.  The  truth  of  Kepler's  Laws  regarding  the  motion  of  a  planet 
would  prove  that  the  force  of  attraction  between  the  planet  and  the 
(fixed)  sun  is  in  the  direction  of  the  line  joining  their  centres, 
and  is  inversely  proportional  to  the  square  of  the  distance  between 
them.  These  laws  are  :  — 

I.  Each  planet  moves  in  an  ellipse  of  which  the  sun  occupies 
one  focus.     (The  path  of  a  comet  may  be  any  conic  section.) 

II.  In  that  ellipse  the  radius-vector  traces  out   equal  areas   in 
equal  times. 

III.  The  square  of  the  periodic  time  is  proportional  to  the  cube 
of  the  mean  distance  between  the  planet  and  the  sun. 

If  equal  areas  are  traced  out  in  equal  times,  the  quantity 


is  zero  (§  50),  which  means  that  the  attraction  is  central,  and  the 
magnitude  of   the   central  attraction  is   (§43)  r-r'02.     Now  the 


GRAVITATION..  109 

equation  of  any  conic  section  referred  to  a  focus  as  pole  is 
r  =  I/a  (1  -f-e  cos  0),  a  being  the  semi-axis  major,  and  e  being  the 
eccentricity.  Hence  (writing  r-Q  =  h,  from  which  we  a't  once  find 


we  easily  obtain  by  the  methods  of  Chap.  V.,  r  —  rt?2  =  —  ah-fr2, 
which  shows  that  the  force  is  attractive  and  varies  inversely  as 
the  square  of  the  distance.  Finally,  since  r20  =  h,  we  have 
a2(l  +  e  cos  9)dO  =  hdt ;  and  the  integral  of  this  throughout  a  com- 
plete revolution,  i.e.,  from  9  =  o  to  9  =  2?r,  is  lit  =  2rf.  But  if  p  be 
the  acceleration  at  the  mean  distance,  a,  so  that  h2  =  pa,  we  find 
4^%3  =  /*a£3.  And  so,  from  the  third  law  of  Kepler,  we  deduce  the 
result  that  gravity  depends  only  on  the  quantity,  and  not  on  the 
quality,  of  matter ;  for  any  two  bodies,  having  the  same  mean 
distance  from  the  sun,  would  have  the  same  periodic  time,  provided 
only  that  their  masses  were  the  same. 

87.  If,  from  the  above  evidence,  we  now  assert  Newton's  great 
Law  of  Gravitation  that  Evert/  particle  of  matter  in  the  universe 
attracts  every  other  particle  with  a  force  whose  direction  is  that 
of  the  line  joining  the  two,  and  whose  magnitude  is  directly  as  the 
product  of  their  masses,  and  inversely  as  the  square  of  their  dis- 
tance from  each  other,  we  see  that  Kepler's  Laws  cannot  be  strictly 
true.     In  the  first  place  all  the  bodies  in  the  solar  system  (including 
the  sun)  will  revolve  about  the  centre  of  inertia  of  the  system, 
which  is  not  necessarily  nor  actually  situated  at  the  sun's  centre  ; 
and  again,  their  paths  cannot  be  true  ellipses  because  of  mutual 
attraction. 

But  we  know  that  Kepler's  Laws  are  not  strictly  true  ;  and, 
further,  the  deviations  from  them  are  precisely  such  as  should 
result  from  mutual  attraction  amongst  the  various  planets.  Indeed, 
by  assuming  the  truth  of  Newton's  Laws  of  Motion  and  of  the  Law 
of  Gravitation,  Adams  and  Leverrier  were  able  to  predict  in- 
dependently the  existence  and  position  of  the  previously  unknown 
planet  Uranus. 

The  law  of  gravitation  is  supported  by  almost  as  strong  proof  as 
any  theoretical  statement  can  possibly  have. 

88.  So  far  as  we  have  gone  we  have  looked  upon  the  sun  and  the 
planets  as  mere  material  points.     The  justification  of  this  is  con- 
tained  in   the    latter   of    two   theorems   due    to    Newton :    1°,   A 
spherical  shell  composed  of  uniform  gravitating  matter  exerts  no 
resultant  attraction  upon  a  particle  in  its  interior ;  and,  2°,  it 


110  A   MANUAL    OF    PHYSICS.' 

attracts  an  external  particle  as  if  its  whole  mass  were  condensed 
at  its  centre. 

Let  A  be  a  point  external  to  the  spherical  shell  PQQ'P',  the 
centre  of  which  is  at  0,  and  the  density  of  which  per  unit  surface 
is  p.  Draw  a  cone,  APQ,  of  infinitely  small  angle  oj,  having  its 
vertex  at  A,  and  intercepting  small  surfaces  of  the  sphere  at  P  and 
Q.  These  surfaces  are  equally  inclined  to  APQ,  and  so  the  masses 
which  are  cut  off  at  P  and  Q,  being  equal  to  wpAP2  sec  OPQ  and 
wpAQ2  sec  OPQ  respectively,  attract  a  particle  at  A  equally.  Now 
take  an  exactly  similar  cone  at  AP'Q',  equally  inclined  to,  but  on 


FIG.  56. 

the  opposite  side  of,  AO.  The  elements  of  the  shell  at  P  and  Q  attract 
the  particle  at  A  equally.  Let  PQ'  intersect  AO  in  R.  The  position 
of  R  is  obviously  independent  of  that  of  P,  and  hence  R  is  the  vertex 
of  a  cone,  of  angle  w',  say,  which  intercepts  the  same  elementary 
areas  of  the  shell  at  P  and  Q'  as  the  cones  APQ  and  AP'Q'  inter- 
cept. The  masses  are  therefore  «/pPR2secOPR  and  w'pQ'R3secOPR 
respectively ;  and,  since  OPR  =  OAP,  their  resultant  attraction 
on  unit  mass  at  A  is 

'PR*  ,  Q'] 


But  PR/PA  =  BR/BA  =  OP/0  A  =  Q'R/Q'A,  and  so  the  resultant  is 


Hence,  summing  all  such  quantities  for  each  similar  pair  of  ele- 
ments, the  whole  attraction  of  the  shell  is  4?rp  OP2/OA2  in  the 
direction  AO.  And  so  the  proposition  is  proved,  since  4;rOP2  is  the 
whole  surface  of  the  shell. 

Since  the  proposition  is  true  of  a  shell,  it  is  also  true  of  a  solid 


GRAVITATION.  Ill 

sphere,  which  may  be  regarded  as  built  up  of  a  number  of  such 
shells.  It  is,  therefore,  practically  true  of  the  planets  and  of  the 
sun ;  for  the  planets,  even  although  in  no  case  strictly  composed 
of  uniformly  dense  concentric  layers,  are  yet  at  distances  from  the 
sun  which  are  large  in  comparison  with  their  own  dimensions. 

[When  A  is  inside  the  shell,  P  and  Q  are  on  opposite  sides  of  it, 
and  so  the  truth  of  the  first  theorem  above  is  established.] 

The  above  case  furnishes  one  example  of  that  limited  class  of 
bodies  which  attract,  and  are  attracted  by,  external  bodies,  as  if 
their  whole  mass  were  condensed  at  a  definite  point,  called  their 


A 


Ef 


FIG.  57. 

centre  of  gravity.  Another  example  is  given  in  §  317.  The  centre 
of  gravity,  -when  it  exists,  always  coincides  with  the  centre  of 
inertia. 

89.  The  second  theorem  is  made  use  of  in  Cavendish's  method  of  de- 
termining the  mass  (and,  consequently,  the  mean  density)  of  the  earth. 

Two  small  leaden  balls,  of  mass  m,  are  attached  to  a  light  rigid 
rod  or  tube,  ab,  which  is  attached  at  its  middle  point  to  a  vertical 
wire.  The  couple  required  to  twist  the  wire  through  a  given  angle 
is  determined  by  observations  upon  the  time  of  oscillation  of  the 
system.  Two  large  leaden  balls  of  mass  M,  which  were  originally 
in  the  positions  A'  B',  are  placed  in  the  positions  A,  B.  The 
mutual  attractions  of  the  balls  A  and  a,  B  and  b,  deflect  ab  from  its 


112  A    MANUAL    OF    PHYSICS. 

normal  position,  and  it  oscillates  about  a  new  position.  The  angle 
through  which  ab  is  deflected  is  determined  by  means  of  a  beam  of 
light  reflected  from  a  mirror  which  is  fastened  to  the  suspending 
wire.  Similar  observations  are  made  with  the  large  balls  once  more 
in  the  positions  A',  B',  and  finally  in  the  positions  A",  B",  at  the 
same  distance  as  before  from  ab,  but  on  opposite  sides  of  them,  so 
as  to  deflect  ab  in  the  opposite  sense.  The  mean  of  the  deflections 
is  taken,  and  the  couple  required  to  produce  it  is  known.  In  this 
way  the  attraction  between  the  two  masses  M  and  m,  their  centres 
being  at  a  given  distance  r  apart,  is  found  ;  and  it  may  be  compared 
with  the  attraction  of  the  earth  upon  the  small  ball.  The  mass  of 
the  earth  being  p,  and  its  radius  E,  we  have  &/t/B3=M/r2.  This 


where  p  is  the  mean  density  of  the  earth,  and  Jf  is  a  constant. 

Professor  C.  V.  Boys  has  recently  succeeded  in  drawing  out 
extremely  fine  fibres  of  quartz,  the  torsional  rigidity  of  which  is 
so  small  that,  by  their  means,  the  mutual  attraction  between  two 
lead  pellets  can  easily  be  made  manifest. 

90.  The  Schehallien  experiment,  undertaken  with  the  object  of 
determining  the  value  of  p,  was  of  precisely  the  same  nature.  The 
deflection  of  the  bob  of  a  pendulum  from  the  true  vertical  under 
the  attraction  of  the  mountain  Schehallien  was  obtained  by 
determining  the  apparent  difference  of  latitude  of  two  places  —  one 
being  situated  to  the  north,  and  the  other  to  the  south  of  the 
mountain  —  by  means  of  a  pendulum  used  as  a  plumb-line.  If 
from  this  we  subtract  the  true  difference  of  latitude  of  the  places, 
we  get  a  measure  of  the  attraction  of  the  mountain  upon  the  bob  of 
the  pendulum  as  compared  with  the  attraction  of  the  earth  upon  it  ; 
for  the  attraction  of  the  mountain  pulls  the  pendulum  from  the 
vertical  in  such  a  way  as  to  increase  the  apparent  latitude  of  the 
northern  point,  and  to  diminish  that  of  the  southern.  The  great 
objection  to  this  method  lies  in  the  fact  that  our  knowledge  of  the 
.mass  of  the  mountain  is  necessarily  very  imperfect. 

Yet  another  method  consists  in  observing  the  times  of  oscillation 
of  pendulums  of  the  same  length,  one  of  which  is  situated  at  the 
top,  and  the  other  at  the  bottom,  of  the  shaft  of  a  coal-pit.  In  this 
way  we  get  a  comparison  of  the  attractions  of  the  whole  earth,  and 
of  the  earth  minus  a  shell  of  thickness  equal  to  the  depth  of  the 
mine,  upon  a  mass  situated  at  known  distances  from  the  centre. 
Here,  again,  uncertainty  arises,  because  of  our  limited  knowledge 
of  the  density  of  the  earth's  crust,  which  must  be  assumed  to  be 


GRAVITATION.  113 

equal  to  that  in  the  neighbourhood  of  the  shaft.  This  is  called  the 
Harton  experiment,  because  it  was  performed  at  the  Harton 
mines. 

The  value  of  the  mean  density  of  the  earth  obtained  by  the 
Schehallien  experiment  is  considerably  less  than,  and  the  value 
obtained  by  the  Harton  method  is  considerably  greater  than,  the 
mean  of  the  (very  closely  accordant)  results  obtained  by  different 
experimenters  who  used  the  Cavendish  method.  Their  results 
show  that  the  mean  density  is  about  5 '5  times  the  density  of 
water. 

Hypotheses  Framed  to  Explain  Gravitation. 

91.  Various  attempts  at  an  explanation  of  gravitation  have  been 
made — all  more  or  less  unsatisfactory. 

One  of  the  most  noted  of  these  is  Le  Sage's  hypothesis  of  ultra- 
mundane corpuscles.     According  to  Le  Sage,  gravitation  is  due  to 
the  bombardment  of  bodies  by  numberless  small  material  particles 
which  are  darting  about  in  space  with  great   speed  in  all   direc- 
tions.    A  single  material  body  placed  in  space  would  not  be  im- 
pelled in  any  one  direction  more  than  in  another,  for  the  corpuscles 
would  batter  it  equally  on  all  sides.     But  two  bodies  in  space  would 
be  driven  together — provided,  at  least,  that  their  distance  apart  were 
small  in  comparison  with  the  free-path  (§  148)  of  the  corpuscles — 
for  those  sides  of  the  bodies  which  face  each  other  would  be  shielded 
to  a  greater  or  less  extent  from  the  bombardment.     If  the  dimen- 
sions of  the  bodies  were  very  much  smaller  than  their  distance 
apart,  the  force   of  attraction   (the  incongruity  of  which  term  is 
evident  from  the  fact  that  the  force  is  here  really  one  of  impulsion) 
would  vary  in  direct  proportion  to  the  cross-sectional  area  of  each 
of  the  two  bodies  as  seen  from  the  other.     But  this  is  not  the 
gravitational    law.      In    order  to  obtain  it,   we  must  make   the 
supposition  that  the  molecules  of  matter  are  so  far  apart  that  the 
number   of    corpuscles  which   pass   completely  through    (say)    the 
earth,  without  striking  any  part  of  it,  is  enormously  larger  than 
the  number  of  those  which  are  stopped  by  it.     In  this  case,  every 
molecule  in  the  interior  will  be  bombarded  equally  with  an  exterior 
particle.     The  force  also  will  be  inversely  as  the  square   of  the 
distance,  and  so  we  get  Newton's  Law. 

In  order  that  the  planets  should  not  experience  appreciable 
resistance  to  their  motion  around  the  sun,  it  is  necessary  to  assume, 
further,  that  the  speed  of  the  planets  is  zero  relatively  to  that  of 
the  corpuscles.  And  this  indicates  one  great  defect  of  the 
hypothesis ;  for  the  energy  of  the  corpuscles  which  must  be  spent 

8 


114  A    MANUAL    OF    PHYSICS. 

in  the  maintenance  of  gravitation  would  be  sufficient,  by  its 
transformation  into  heat,  to  completely  volatilise  any  material 
substance  of  which  we  have  knowledge. 

92.  The  law  of  gravitation  may  be  worked  out  into  all  its  conse- 
quences (at  least,  so  far  as  our  methods  avail)  without  any  know- 
ledge of  the  mechanism  by  which  it  occurs.     We  require  merely  to 
assume    that    it    acts    directly   at   a   distance.      But,   as   Newton 
remarked,  no  one  competent  to  think  correctly  on  physical  matters 
will  be  content  with  this  assumption.     He  himself  suggested  the 
rarefaction  of  the  ether  in  the  neighbourhood  of  dense  bodies  as  a 
possible  explanation.     Sir  W.  Thomson  has  pointed  out  a  dynamical 
method  of  producing  this  diminution  of  pressure.    An  incompressible 
fluid,  filling  all  space,  which  is  brought  into  existence,  or  is  annihi- 
lated, at  the  surface  of  every  particle  of  matter  at  a  rate  propor- 
tioned to  the  mass  of  that  particle,  and  which  is  annihilated,   or 
produced,  at  an  infinite  distance,  at  the  same  total  rate  would  supply 
the  necessary  means.    .(See  §  74.) 

Waves  traversing  a  medium  would  have  the  effect  of  making 
bodies  immersed  in  it  approach  each  other. 

The  property  of  dilatancy  (Chap.  XXXIII.)  in  a  medium  com- 
posed of  rigid  particles  in  mutual  contact  would  also  account  for  a 
gravitational  action  on  bodies  placed  in  it. 

A  certain  stress  in  Maxwell's  electro-magnetic  medium  (Chap. 
XXXII.)  would  account  for  it  too.  So,  as  we  have  seen,  would 
Le  Sage's  ultra-mundane  corpuscles.  But,  to  every  provisional 
hypothesis  yet  brought  forward,  some  objection,  more  or  less  con- 
clusive, may  be  advanced. 

If  gravitation  be  due  to  action  propagated  through  a  material 
medium,  its  propagation  through  finite  distances  must  occupy  a 
finite  time.  We  can  assert  merely  that  the  speed  of  propagation 
is  large  in  comparison  with  planetary  velocities  ;  for  no  such  modifi- 
cation of  the  planetary  motions,  as  would  be  entailed  were  it  other- 
wise, is  observable. 

The  Nebular  Hypothesis. 

93.  Various  nebulae,  when  examined  under  sufficiently  high  mag- 
nifying power,  are  seen  to  be  merely  groups  of  stars  like  our  own 
stellar  system.     But  others   cannot   be  so  resolved ;    and,   among 
them,  there  is  great  variety  of  constitution.     Some  appear  to  be 
comparatively   uniform    throughout,   while    others    seem   to   vary 
greatly  in  density  at  different  parts.     Others,  again,  have  very  dense 
nuclei  with  a  faint  nebulous  surrounding. 

Such  facts  as  these  suggested  to  Laplace  his   famous   Nebulae 


GRAVITATION.  115 

Hypothesis.  According  to  this  hypothesis,  we  see  in  these  nebulae 
solar  systems,  such  as  our  own,  in  various  stages  of  formation.  The 
sun  was  formed  by  the  mutual  gravitation  of  its  parts  from  a 
state  of  diffusion  throughout  space.  And,  to  account  for  its  rotation, 
these  parts  must  have  had  the  same  amount  of  moment  of  momen- 
tum about  an  axis  as  that  which  the  sun  at  present  has.  In  all 
probability  they  had  small  translational  velocity  before  they  began 
to  gravitate  towards  each  other  —  small,  that  is,  relatively  to  the 
speed  which  they  would  acquire  under  the  action  of  gravitation  ;  for, 
if  this  were  not  so  the  chance  of  their  colliding  would  be  vanishingly 
small.  The  collision  of  these  parts  would  produce  great  heat,  which 
would  result  in  the  production  of  a  nebula,  extending  beyond  the 
orbit  of  Neptune,  and  slowly  revolving  on  its  axis.  As  the  nebula 
gradually  shrank,  its  angular  velocity  would  increase  until,  by  '  centri- 
fugal force,'  a  ring  of  matter  was  left  behind  as  the  body  of  the  nebula 
still  farther  shrank.  The  breaking  up  of  this  ring  (which  would 
usually  occur,  as  dynamical  principles  show),  and  the  subsequent 
agglomeration  of  its  parts,  would  result  in  the  formation  of  a  planet. 
And  so  the  development  would  proceed. 

Laplace's  hypothesis  cannot,  as  he  originally  stated  it,  give  a,  full 
account  of  all  the  phenomena  of  our  solar  system  ;  but  this  we  do 
know,  that  some  such  hypothesis  must  be  true  if  our  sun's  heat  has 
had  a  physical  origin  such  as  our  present  knowledge  is  adequate  to 
explain.  We  cannot  conceive  of  any  store  of  energy,  sufficient 
to  account  for  the  immense  radiation  of  heat  from  the  sun,  except 
the  potential  energy  of  separated  masses  of  gravitating  matter. 

Potential. 

94.  Since  any  force  /  is  equal  to  the  space-rate  of  variation  of 
the  kinetic  energy  of  the  material  system  upon  which  it  acts,  and 
since,  by  the  principle  of  conservation,  the  change  of  kinetic  energy 
is  equal  and  opposite  to  the  change  of  potential  energy,  we  may 

write  ,de         dV 


where  V  represents  the  potential  energy. 

The  value  of  the  quantity  V  depends  upon  the  instantaneous 
position  of  the  material  system,  and  also  upon  the  mutual  configura- 
tion of  its  various  parts. 

We  may  define  the  mutual  potential  energy  of  two  material 
bodies  in  any  given  relative  position,  as  the  amount  of  work  which 
may  be  obtained  by  allowing  them  to  move,  under  their  mutual 
repulsion,  to  an  infinite  distance  apart.  And,  of  course,  this 

8—2 


116  A   MANUAL    OF   PHYSICS. 

definition  implies  that  we  choose  the  configuration  of  infinite  distance 
as  the  configuration  of  zero  potential  energy. 

The  Potential  at  any  point,  due  to  any  given  distribution  of 
matter,  is  the  mutual  potential  energy  between  that  matter  and 
a  unit  of  matter  placed  at  the  given  point.  To  find  its  value 
we  have  only  to  multiply  the  force  at  any  point  of  the  path,  along 
which  the  unit  of  matter  is  repelled,  by  the  infinitesimal  element, 
ds,  of  the  path,  and  to  sum  all  such  quantities  from  the  given  posi- 
tion to  an  infinite  distance.  This  quantity  is  represented  by  the 
symbol  « 


the  meaning  of  which  is  that  we  are  to  find  the  general  value  of  the 
integral  oifds  ;  to  replace  in  it,  first,  s  by  oo  ;  second,  s  by  its  actual 
value  at  the  given  point ;  and,  finally,  to  subtract  the  latter  quan- 
tity, so  found,  from  the  former.  This  makes  the  work  which  is 
done  depend  only  upon  the  initial  and  the  final  positions  of  the  unit 
of  matter — a  condition  which  must  be  satisfied  if  the  forces  are 
consistent  with  the  principle  of  conservation.  For,  if  the  work 
depended  upon  the  path  along  which  the  matter  was  repelled,  we 
might  cause  it  to  return,  by  frictionless  constraint,  to  its  former 
position,  by  a  path  which  necessitates  an  expenditure  of  less  work 
than  that  which  was  done  upon  it  by  repelling  forces.  In  such  a 
case  there  would  consequently  be  a  continual  gain  of  energy  without 
any  corresponding  expenditure. 

Let  us  suppose,  for  example,  that  the  mutual  force  is  repulsive 
and  inversely  proportional  to  the  square  of  the  distance,  s,  between 
the  two  portions  of  matter.  Let  s  increase  from  s  to  -s'.  The  work 
which  is  done  is 


where  m  is  the  mass  of  the  repelling  matter,  and  we  assume  that 
the  law  of  repulsion  is  similar  to  Newton's  law  of  attraction,  and 
define  unit  force  as  the  force  between  two  unit  masses  placed  at  unit 
distance  apart.  The  general  value  of  the  integral  is  (§  35,  example  (3) ) 


and  the  work  is  therefore 


GRAVITATION.  117 

If  we  now  put  s'  =  oo  ,  the  second  term  disappears  and  we  see  that 
the  value  of  the  potential  is 


and  the  repulsive  force  is 


95.  Gravitational  Potential.  —  When  the  force  is  attractive,  as  in 
the  case  of  gravitation,  the  potential  —  denned  as  above  —  becomes 

V=--- 

s 

The  potential  is  therefore  essentially  negative  ;  that  is  to  say,  work 
is  done  against  the  forces  when  the  distance  between  the  particles  is 
increased.  But  it  must  carefully  be  observed  that  the  fact  of  the 
potential  energy  being  negative,  at  distances  less  than  infinite,  is 
due  entirely  to  our  having,  for  convenience,  chosen  infinite  distance 
apart  as  the  configuration  of  zero  potential.  The  increase  of 
potential  energy  is  positive  as  we  pass  towards  infinity.  Still,  to 
avoid  the  inconvenience  of  defining  the  gravitational  potential  as  a 
negative  quantity  at  all  finite  distances,  it  is  preferable  to  change 
the  sign  and  write,  in  this  case  also, 

y_  m 
s 
dV 


where  /  is  to  be  understood  as  signifying  inward  or  attractive  force. 
The  potential  V,  therefore,  does  not  represent  the  mutual  potential 
energy.  It  is  the  exhaustion  of  potential  energy  which  occurs  when 
the  unit  of  matter  passes  from  infinity,  under  the  attractive  forces, 
to  the  position  s. 

The  potential  at  any  point,  due  to  a  number  of  separate  masses, 
is  simply  the  sum  of  the  separate  potentials  due  to  each  mass. 

96.  Equipotential  Surfaces.  Lines  and  Tubes  of  Force.  —  Any 
surface  over  which  V  is  constant  is  called  an  Equipotential  Surface  ; 
and  a  line,  the  tangent  to  which  at  any  point  is  always  in  the 
direction  of  the  force  at  that  point,  is  called  a  Line  of  Force.  Each 
line  of  force  is  a  possible  path  along  which  a  material  particle  would 
move  under  the  given  forces. 

Since  dV  is  zero  as  we  pass  from  one  point  to  another  point  of  an 
equipotential  surface,  it  follows  that  the  force  at  any  point  of  such 
a  surface  has  no  component  along  it.  In  other  words,  the  lines  of 
force  are  everywhere  perpendicular  to  the  equipotential  surfaces. 


118  A   MANUAL   OF   PHYSICS., 

No  two  different  equipotential  surfaces  can  intersect  with  one 
another  ;  for  this  would  imply  that  a  finite  change  of  potential  would 
follow  an  infinitely  small  displacement  of  a  material  particle,  i.e., 
the  force  would  be  infinite  at  any  such  intersection,  and  no  examples 
of  infinite  forces  occur  in  nature. 

If,  through  every  point  of  an  infinitely  small  closed  curve  drawn 
on  an  equipotential  surface,  we  draw  lines  of  force,  a  tube  of 
infinitely  small  section  will  be  formed.  Such  a  tube  is  called  a 
Tube  of  Force. 

Necessarily,  the  lines  and  tubes  of  force  can  originate  only  at 
a  point  where  matter  is  situated  ;  for  the  force  owes  its  origin  to 
the  presence  of  matter.  And  if,  at  any  point  of  space,  a  force 
f  exists,  we  may  draw  an  infinitesimal  tube  of  force,  so  as  to  con- 
tain that  point  and  to  enclose  f  lines  of  force  per  unit  area  of  its 
normal  section.  The  number  of  the  lines  of  force  per  unit  of 
sectional  area  therefore  indicate  the  intensity  of  the  force  at  the 
given  point. 

If  a  given  portion  of  a  tube  does  not  contain  any  matter,  the 
number  of  lines  which  it  contains  remains  constant,  since  no  line  can 
end  in  its  interior,  and  none  can  pass  out  through  its  sides.  This  gives 


where  f  is  the  force  at  any  point  of  the  tube  at  which  the  sectional 
area  is  a,  and  c  is  a  constant,  numerically  equal  to  the  number  of 
lines  which  the  tube  contains. 

97.  Special  Applications.  —  (1)  Let  the  attracting  body  be  sym- 
metrical about  a  point.  In  this  case  the  tubes  of  force  are  cones, 
and  the  area  of  any  normal  section  of  each  of  them  is  proportional  to 
the  square  of  its  distance  from  the  point  of  symmetry.  Hence  the 
above  equation  shows  that  the  force  at  any  point  is  inversely  pro- 
portional to  the  square  of  the  distance  of  that  point  from  the  point 
of  symmetry,  say 


(2)  Let  the  attracting  matter  be  symmetrically  arranged  about  an 
infinitely  long  straight  axis.  The  equipotential  surfaces  are  concentric 
cylinders  whose  common  axis  coincides  with  the  axis  of  symmetry, 
and  the  tubes  of  force  are  wedges  bounded  by  axial  planes.  The 
section  is  proportional  to  the  distance  from  the  axis,  and  therefore 
the  force  is  inversely  proportional  to  that  distance,  i.e., 


GRAVITATION.  119 

(3)  Let  the  matter  be  arranged  homogeneously  in  infinite  parallel 
planes.  The  equipotential  surfaces  are  planes  parallel  to  these,  and 
the  tubes  of  force  are  cylinders  arranged  perpendicularly  to  the 
planes.  Consequently  the  force  is  constant  at  all  distances,  say 


98.  Total  Force  over  a  Closed  Surface.  —  Draw  any  closed 
surface  S  (Fig.  58),  and, 

(1)  Let  m  be  a  massive  particle,  outside  the  surface,  to  which 
the  force  is  due.  Draw  any  infinitesimal  tube  of  force,  mnp,  cutting 
the  surface  at  n  and^?.  (Of  course,  it  may  cut  the  surface  in  any 
even  number  of  places.)  The  total  force  over  the  portion  of  the 
surface  intercepted  by  the  tube  at  n,  is  equal  to  that  over  the 
portion  intercepted  at  p  ;  but,  in  the  one  case,  the  force  is  directed 
outwards  over  the  surface,  while,  in  the  other,  it  is  directed 
inwards.  Hence,  by  consideration  of  an  infinite  number  of  such 


FIG.  58. 

tubes  of  force  intersecting  all  parts  of  the  closed  surface,  we  see 
that  the  total  inward  force  over  the  whole  closed  surface,  due  to  a 
material  particle  situated  at  an  external  point,  is  zero.  [The  use 
of  the  word  inward,  of  course,  implies  that  the  force  is  to  be 
reckoned  as  negative  when  it  is  outwardly  directed.] 

The  same  proof  applies  in  the  case  of  any  number  of  material 
particles. 

(2)  Let  the  particle,  of  mass  m,  be  placed  within  the  closed 
surface.  Draw  a  sphere,  with  unit  radius,  from  the  point  m  as 
centre.  The  area  of  this  spherical  surface  is  47r,  and  the  force  at 
any  point  of  it,  due  to  the  attraction  of  the  central  particle,  is 
equal  to  m.  The  total  inward  force  is  therefore  4-n-m.  But,  by  the 
result  of  §  96,  the  total  inward  force  over  this  surface  is  equal  to 
that  over  the  given  surface  S.  Hence,  if  m  be  the  whole  amount 
of  matter  contained  within  S,  the  total  inward  force  over  the  whole 
closed  surface,  due  to  matter  of  amount  m  enclosed  within  it,  is 


120  A  MANUAL   OF   PHYSICS. 

These  results  may  be  symbolised  thus, 
/NdS  =  0  or  4tirm, 

where  N  rep  esents  the  number  of  lines  of  force  which  cross  the 
surface  S  per  unit  of  area  in  the  part  where  the  element  of  surface 
dS  is  taken. 

99.  Special  Applications.  —  We  may  apply  these  results  to  the 
determination  of  the  values  of  the  constants  a,  b,  and  c,  in  §  97. 

In  example  (1)  of  that  section,  the  whole  number  of  lines  of 
force  which  cross  any  closed  surface  surrounding  the  spherical 
distribution  of  attracting  matter  is  47rm,  where  m  is  the  whole 
amount  of  matter.  Hence,  if  we  suppose  the  enclosing  surface  to 
be  a  concentric  sphere  of  radius  s,  we  get  as  the  total  force 


Therefore 


Similarly,  if,  in  example  (2),  we  consider  the  force  due  to  the 
matter  m  contained  in  unit  of  length  of  the  cylindrical  distribution, 
we  get 


for  27TS  is  the  area  of  unit  of  length  of  a  concentric  cylinder  of 
radius  s. 

Therefore  b  =  2m. 

Again,  if  (example  (3)  )  we  draw  a  right  circular  cylinder  of  unit 
radius,  perpendicular  to  the  infinite  planes,  and  close  its  ends  by 
parallel  planes  on  opposite  sides  of  the  given  plane  distribution  of 
matter,  the  force  exerted  over  the  two  ends  is 


27r/=  2?rc  = 
so  that  c  =  2w. 

In  particular,  if  the  given  distribution  consists  of  an  infinitely 
thin  plane  layer,  of  infinite  extent,  and  of  finite  surface  density  a, 
the  force  at  any  point  is 

/  =  2m  =  2™. 

[Such  a  distribution  cannot  occur  in  the  case  of  gravitational 
matter,  but  the  problem  has  a  direct  application  in  the  theory  of 
electricity.]  It  follows  that  the  force  at  any  point  just  outside  a 
surface  on  which  matter  is  distributed  is  normal  to  the  surface  and 


GRAVITATION.  121 

equal  to  2;r(r  —  provided  that  the  surface  is  of  finite  curvature  ;  for 
the  point  may  be  taken  so  close  to  the  surface  that,  as  seen  from  it, 
the  surface  is  practically  an  infinite  plane  ;  that  is  to  say,  any 
infinitely  small  portion  of  the  surface  is  plane,  and  the  point  may 
be  taken  infinitely  close  to  this  portion  in  comparison  with  its 
dimensions,  infinitely  small  though  they  be. 

.  100.  In  the  case  just  considered,  the  normal  force  at  the  two 
sides  of  the  surface  are  respectively  /=  27r<r,  and/'  =  -  27nr.  The 
total  change  of  force  in  crossing  the  surface  is  therefore 


This  shows  us  how  to  distribute  matter  over  a  given  surface  in 
order  to  produce  a  given  change  in  the  value  of  the  normal  force 
in  passing  from  one  side  to  the  'other. 

We  may  write  this  expression  in  the  form 


where  V  and  V  are  the  values  of  the  potential  at  each  side  of  the 
surface,  while  dn  and  dn'  are  measured  along  the  outwardly  drawn 
normals  on  each  side.  This  enables  us  to  calculate  the  distribution 
of  matter  when  the  discontinuous  distribution  of  potential  is 
given. 

It  is  easy  also  to  obtain  an  expression  for  the  volume  -density  of 
matter  which  is  required  to  produce  a  given  continuous  distribution 
of  potential. 

For,  since  the  force  /outside  a  symmetrical  spherical  distribution 
of  matter  is  (§  99)  given  by  the  equation 


the  value  of  /  just  outside  the  sphere  is  /  =  m/r2  (which,  we  may 
observe  in  passing,  proves  Newton's  theorem,  §  88).  But  m  is 
equal  to  4/3  .  Trpr3,  if  the  sphere  is  of  uniform  density  p  ;  in  which 
case,  therefore,  /=  4/3  .  ?rpr.  That  is 

dV      4 


whence  -  V  =  ?7rpr2  +  C  (a  constant)  . 

3 

Now,  if  we  take  the  origin  of  co-ordinates  at  the  centre  of  the 
sphere,  we  have 

>•-•  =  3?  +  7/2  +  zi  . 

whence  drjdx  =  xjr,  dr/dy  =  yjr,  drjdz  =  z/r,  for,  since  x,  y,  and  £, 


122 


A   MANUAL    OF    PHYSICS. 


are  independent  variables  (§  28),  we  must  assume  y  and  z  to  be 
constant  when  x  varies,  and  so  on.     Therefore  we  get 
dV4       dr4 


#V=4 


And,  similarly,  -  — 1-  =   ; 


whence 


4 


Around  any  point  in  space  throughout  which  matter  is  distributed 
with  density  p,  describe  a  sphere  which  is  so  small  that  the  density 
of  the  matter  which  it'  contains  is  sensibly  constant.  We  may 
suppose  all  the  attracting  matter  to  consist  of  two  portions — that 
which  is  within  the  little  sphere,  and  that  which  is  external  to  it. 
We  may  divide  the  whole  potential  V  into  two  parts,  Vi  and  V2, 
of  which  the  former  is  due  to  the  sphere  and  the  latter  is  due  to  the 
matter  external  to  the  sphere.  Vi  therefore  satisfies  the  equation 


p  being  the  density  of  the  matter  which  produces  the  potential  Vi 
at  the  given  point.  But  the  matter  outside  the  sphere  does  not 
contribute  to  the  density  at  the  given  point  within.  That  is,  at  the 
given  point,  the  density  of  the  matter  which  produces  the  potential 
V2  is  zero,  and  therefore 


\dx'2 


+  -VV    = 


FIG.  59. 

Consequently,  by  addition,  we  get  quite  generally 

d*V  +  d^\_ 
+~*~ 


GRAVITATION.  123 

This  shows  us  how  to  distribute  matter  throughout  space,  so  as  to 
produce  a  given  continuous  distribution  of  potential. 

The  above  result  may  be  obtained  in  a  totally  different  manner, 
which  is  even  simpler,  and  which  will  make  the  meaning  of  the 
equation  more  evident. 

Take  three  rectangular  axes  at  any  point  o  (Fig.  59),  and  draw  a 
little  parallelepiped  at  this  point  with  its  edges  parallel  to  the  a?,  y, 
and  z  axes ;  and  let  the  lengths  of  the  edges  be  8x,  dy,  Sz,  respectively. 

Eesolve  the  forces  in  the  neighbourhood  of  the  parallelepiped 
into  their  components  parallel  to  the  axes.  This  will  not  alter  the 
result  of  §  98  ;  and  we  may  now  draw  the  lines  of  force  perpendicular 
to  the  various  small  surfaces. 

Let  nf  be  the  number  of  lines  of  force  which  cross  unit  area  of 
the  face  of  the  parallelepiped  which  passes  through  the  origin  and 
is  perpendicular  to  the  x  axis.  The  total  number  of  lines  which 
cross  that  face  is  therefore  n^yd*,  since  Sydz  is  the  area  of  the 
face.  Similarly  the  number  which  cross  the  parallel  face  at  the 
distance  dx  from  the  former  is 


Now  the  lines  which  cross  the  former  face  are  due  entirely  to  matter 
outside  the  little  volume,  and  they  therefore  cross  the  parallel  face 
also.  Hence  the  total  number  of  lines  which  enter  the  little 
volume  from  without  by  the  two  faces  is 


the  difference  of  these  two  quantities.  By  similar  reasoning  for 
the  other  pairs  of  faces  we  find  that  the  total  number  of  lines  which 
enter  the  little  volume  (that  is,  the  excess  of  those  which  enter  over 
those  which  leave)  is 

(dnx  +  dny  +  dn. 

\  dx       dy        dz 


But,  by  §  98,  this  is  equal  to  ^TrpdxSySz,  where  p  is  the  density  of 
the  matter  contained  in  the  volume  dttfyfy.  Therefore 

dnr  .  dnv    .  dnx      A 

-^+-4+-^=^- 

And  the  result  is  independent  of  the  size  of  the  little  volume ;  so 
that  the  meaning  of  the  equation  is  that  the  volume  density  at  any 
point  of  space  is  l/4w  times  the  number  of  lines  of  force  which 
originate,  per  unit  of  volume,  at  that  point. 


124  A  MANUAL   OF   PHYSICS.      , 

The  following  proposition  is  also  of  great  importance.  It  is 
possible  so  to  distribute  matter  over  a  given  surf  ace,  ivhich  encloses 
a  given  mass,  as  to  produce,  outside  that  surface,  the  same  potential 
as  that  which  the  given  mass  produces.  The  mathematical  proof  of 
this  proposition  cannot  be  introduced  here,  but  an  experimental  proof 
will  be  given  in  the  chapter  on  electrostatics. 

101.  The  calculation  of  the  distribution  of  potential  which  is 
produced  by  a  given  distribution  of  matter  is  extremely  difficult  or 
even  impossible  in  most  cases  ;  but  the  beautiful  method  of  electric 
images,  due  to  Sir  W.  Thomson,  enables  us  to  deduce  with  great 
ease  the  solution  of  many  unknown  problems  from  the  known 
solution  of  others.  The  further  discussion  of  this  subject  may  be 
left  until  we  treat  of  the  subject  of  electricity. 


CHAPTEB  IX. 

PROPERTIES    OF    GA-SES. 

102.  Compressibility. — Throughout  this  investigation  we  assume 
that  the  temperature  of  the  gas  remains  constant.  The  effects  which 
result  from  changes  of  temperature  will  be  more  conveniently  treated 
in  the  chapter  on  the  effects  of  heat. 

All  gases  are  compressible ;  that  is,  their  volume  can  be  diminished 
by  the  application  of  pressure.  We  shall  see  afterwards  that  sound 
could  not  pass  at  a  finite  rate  through  a  gas  which  was  not  compres- 
sible. So  that  the  mere  fact  that  gases  can  convey  sound  constitutes 
a  proof  of  their  compressibility. 

103.  Boyle's  Law. — The  law  which  very  completely,  though  not 
with  absolute  accuracy,  represents  the  relation  between  the  pressure 
and  the  volume  of  air  (and  many  other  gases)  was  discovered 
experimentally  by  Boyle,  who  showed  that  the  density  of  a  gas  is 
directly  proportional  to  the  pressure.  In  symbols,  p  being  the 
density  and  p  the  pressure,  this  is 

p  =  cpt 

c  being  a  constant.  The  density,  that  is  the  mass  or  quantity  of 
matter  in  unit  volume,  is  numerically  equal  to  the  reciprocal  of  the 
volume  containing  unit  mass.  Hence,  v  being  this  volume,  we 
may  write,  instead  of  the  above, 

pv  =  c, 

where  the  constant  c  is  the  reciprocal  of  the  former  one;  or,  in 
words,  the  volume  is  inversely  proportional  to  the  pressure. 

Boyle's  apparatus  consisted  of  a  glass  U-tube  (Fig.  60)  with  a  long 
and  a  short  limb.  The  long  limb  was  open  to  the  atmosphere,  while 
the  short  one  was  closed,  and  contained  a  quantity  of  air,  which  was 
separated,  by  means  of  mercury  filling  the  bend  of  the  tube,  from 
the  outside  air.  The  level  of  the  mercury  was  the  same  in  both 
limbs  of  the  tube,  and  so  the  enclosed  air  (the  volume  of  which  was 
carefully  noted)  was  at  atmospheric  pressure  (§75).  Mercury  was 
then  poured  into  the  open  limb,  until  a  difference  of  level  equal  to 


126 


A    MANUAL    OF    PHYSICS. 


a. 


the  height  of  the  mercury  barometer  was  established.  The  air 
inside  was  therefore  under  a  pressure  of  two  atmospheres,  and  its 
volume  was  found  to  have  been  halved  ;  and  so  on  with  other  values 
of  the  pressure. 

A  slight  modification  of  the.  apparatus  enables  us  to  prove  the  law 
under  diminished  pressure.  AB  (Fig.  61) 
is  a  vessel  containing  mercury.  The  glass 
tube  a&,  which  is  closed  at  the  end  a,  but 
is  open  at  6,  is  filled  with  mercury,  and 
inverted  in  AB.  Being  shorter  than  the 
height  of  the  mercury  barometer,  the  tube 
ab  remains  filled.  Air  or  any  other  gas 
may  now  be  introduced  into  it  until  the 
mercury  inside  is  at  the  same  level  as  that 
outside.  Under  these  conditions  the  gas 
in  ab  is  under  atmospheric  pressure.  If 
ab  be  raised,  the  mercury  in  it  stands  at 
a  higher  level  than  that  outside,  and  the 
gas  expands,  since  it  is  under  diminished  *^jj  g 

pressure.      The  ideal  gas  which  rigidly 
obeys  Boyle's  Law  is  called  a  perfect  gas.    FIG.  60.  FIG.  61. 

104.  Compressibility  of  a  Perfect  Gas.  —  One  gas  is  more  compres- 
sible than  another  in  direct  proportion  to  the  alteration  of  volume 
produced  by  a  given  pressure,  and  in  inverse  proportion  to  the 
pressure  required  to  change  the  volume  to  a  given  extent.  Hence 
we  measure  the  compressibility  by  the  ratio  of  the  percentage 
change  of  volume  to  the  change  of  pressure  which  produces  it. 
That  is  to  say,  if  the  volume  V  changes  by  the  quantity  v  when  the 
pressure  alters  by  the  amount  p,  the  compressibility  is  measured  by 
the  ratio  vfVp. 

By  Boyle's  Law  we  have 


and 

Therefore  pV  -  ~Pv  =  0, 

since  we  can  neglect  pi),  which  is  the  product  of  two  small  quantities. 

This  gives  v      1 


that  is,  the  compressibility  of  a  perfect  gas  is  inversely  proportional 
to  the  pressure. 

105.  Deviations  from  Boyle's  Laiv.  —  Though  no  gas  is  perfect, 
yet  many  gases  do  not  greatly  deviate  from  Boyle's  Law  throughout 
a  considerable  range  of  pressure. 

Air  is  more  compressed  than  it  should  be  in  accordance  with  the 


PROPERTIES    OF    GASES.  127 

law  until  a  pressure  of  nearly  one  ton's  weight  on  the  square  inch 
is  reached.  After  this  point,  the  compression  is  less  than  the 
calculated  value.  A  reason  for  this  is  simply  that  the  volume  of  the 
gas  is  not  capable  of  indefinite  decrease,  while  Boyle's  Law  asserts 
that  under  infinite  pressure  the  volume  will  become  zero. 

Hydrogen,  unlike  air,  is,  at  ordinary  temperatures,  always  less 
compressible  than  the  law  indicates.  Nitrogen,  along  with  many 
other  gases,  resembles  air. 

These  results  are  exhibited  graphically  in  Fig.  62.     The  actual 


[FiG.  62. 

volume  of  the  gas  is  measured  along  the  vertical  axis,  while  the 
volume  of  a  perfect  gas  under  the  same  pressure  is  measured  along 
the  horizontal  axis.  The  straight  line  passing  upwards  through 
the  origin  at  an  angle  of  45°  is  obviously  the  graph  for  a  perfect 


128 


A    MANUAL    OF    PHYSICS. 


PROPERTIES   OF   GASES.  129 

gas.  "The  curved  line  intersecting  the  perfect  gas-line,  and,  like  it, 
sloping  upwards  towards  the  right,  represents  the  action  of  air  ; 
and  the  other  curve,  sloping  similar!}-,  is  the  graph  for  hydrogen. 

106.  Compression  of  Vapours. — A  vapour,  though  it  may  obey 
Boyle's  Law  throughout  a  considerable  range  of  pressure,  ulti- 
mately deviates  more  and  more  from  that  law  as  the  pressure  rises. 
The  direction  of  the  deviation  is  similar  to  that  of  air  at  pressures 
less  than  152  atmospheres,  i.e.,  the  vapour  is  more  compressed 
than  a  perfect  gas  would  be.  When  the  pressure  has  become 
sufficiently  great,  the  vapour  begins  to  liquefy ;  and  the  pressure 
then  remains  constant  until  the  whole  has  become  liquid.  Further 
compression  is  comparatively  a  matter  of  extreme  difficulty. 

The  whole  process  above  described  must  take  place  with  extreme 
slowness  in  order  that  the  condition  (§  102)  of  constant  temperature 
may  be  adhered  to. 

We  may  now  suppose  the  temperature  to  be  increased  to,  and 
maintained  at,  a  definite  value  higher  than  that  which  it  formerly  had. 
If  the  pressure  has  the  same  value  as  it  had  at  the  commencement 
of  the  former  process,  the  volume  of  the  vapour  will  be  greater 
than  before,  for  all  vapours  expand  when  heated  under  constant 
pressure.  And,  if  the  pressure  be  increased  as  in  the  previous  case, 
a  precisely  similar  series  of  phenomena  will  be  presented;  the 
volume  of  the  substance,  however,  being  always  larger  than  formerly 
under  the  same  conditions  as  regards  pressure.  But  one  important 
difference  will  be  noted — the  change  of  volume  during  the  process 
of  liquefaction  will  be  less  than  it  was  when  the  temperature  was 
lower.  Ultimately,  when  the  temperature  is  sufficiently  high, 
there  is  no  sudden  change  of  volume  when  the  substance  assumes 
the  liquid  condition.  At  still  higher  temperatures,  the  deviations 
from  Boyle's  Law  become  less  and  less  marked. 

At  all  temperatures  above  the  limiting  one  at  which  sudden  lique- 
faction ceases,  the  substance  is  called  a  gas ;  at  lower  temperatures 
it  is  termed  a  vapour. 

With  this  explanation,  no  difficulty  will  be  experienced  in  under- 
standing the  diagram  on  the  opposite  page.  The  volume,  v,  of  a  per- 
fect gas  is  measured  along  the  horizontal  axis  from  a  point  not  shown 
in  the  diagram.  The  scale  is  such  that  1,000  times  the  reciprocal  of 
the  abscissa  represents  the  pressure  in  atmospheres.  The  actual 
volume,  v',  of  carbonic  acid  gas  is  measured  along  the  vertical  axis. 
In  this  way  a  series  of  curves  are  shown  which  indicate  the  devia- 
tion of  that  gas  from  Boyle's  Law  at  various  temperatures.  Portions 
of  two  of  the  nearly  straight  lines,  which  these  curves  would  become 
if  the  gas  were  air,  are  drawn.  The  vertical  portions  of  two  of  the 

9 


130  A   MANUAL   OF    PHYSICS. 

curves  indicate  the   stage   during  which  liquefaction  occurs — the 
almost  horizontal  parts  belong  to  the  liquid  carbonic  acid. 

The  substance  is  more  or  less  compressible  under  given  conditions 
of  pressure  and  temperature  than  a  perfect  gas  is,  according  as  a 
line  touching  the  curve  (at  the  point  satisfying  these  conditions) 
makes  a  greater  or  less  angle  with  the  horizontal  axis  than  the 
angle  whose  tangent  is  v'jv  times  the  tangent  of  the  angle  made  by 
the  corresponding  perfect-gas  line.  Hence  the  substance,  when  in  a 
condition  resembling  the  liquid  state,  is  less  compressible  than  a 
perfect  gas  would  be ;  and  we  thus  see  that  hydrogen  is  less  com- 
pressible than  a  perfect  gas,  because,  under  ordinary  conditions  of 
temperature  and  pressure,  it  is  in  a  state  more  analogous  to  that  of 
liquids  than  to  that  of  gases.  Were  it  examined  under  conditions 
similar  to  those  of  carbonic  acid  in  the  upper  right-hand  region  of 
the  diagram  opposite,  i.e.,  under  sufficiently  diminished  temperature 
and  pressure,  it  would  almost  certainly  be  found  to  have  a  smaller 
volume  than  Boyle's  Law  shows. 

107.  Elasticity. — All  gases  and  vapours  possess  perfect  elasticity 
of  bulk.     That  is  to  say,  they  entirely  recover  their  original  bulk 
when  allowed  to  do  so  by  means  of  the  removal  of  the  distorting 
pressure.     This  may  readily  be  proved  by  the  simple  experiment  of 
inverting  in  water  a  glass  vessel  containing  air  or  any  gas  which  is 
not  appreciably  dissolved  by  the  liquid.     The  gas  may  be  subjected 
to,  or  relieved  from,  pressure  by  raising  or  lowering  the  glass  vessel. 
The  possibility  of  discharging  a  bullet  from  a  gun,  or  of  propelling 
a  vessel  or   driving  machinery  by  means    of    compressed   gases 
furnishes  another  proof  of  their  elasticity.     And  still  another  proof 
consists  in  the  fact  that  they  all  convey  sound,  which  would  be 
impossible  were  they  not  elastic,  just  as  it  would  be  impossible  if 
they  were  not  capable  of  being  compressed. 

108.  Viscosity. — We  have  already  given  a  general  definition  of 
this  term  as  the  property  in  virtue  of  which  there  is  resistance  to 
shearing  motion.     But  it  is  convenient  to  use  the  word  as  referring 
to  a  specific  property  (one  independent  of   the    size  of   the  body, 
§  81).     Hence  we  define  viscosity  as  the  tangential  force  per  unit 
area  of  two  indefinitely  large  parallel  plane  surfaces  of  the   fluid 
which  are  at  unit  distance  apart  and  move  parallel  to  each  other  with 
unit  relative  speed.      It  follows  that  the  tangential  force  per  unit 
area  of  two  such  planes  at  a  distance  x  apart,  and  moving  with 
relative  speed  v,  is  rv/x,  where  T  is  the  viscosity.     But,  in  shearing 
motion  v  is  always  proportional  to  x,  so  that  the  tangential  force 
is  rdvldx. 

In  making  an  actual  determination  of  the  value  of  r  in  any  gas, 


PROPERTIES    OF   GASES.  131 

various  forms  of  experiment  based  upon  the  above  definition  might 
be  used.  Clerk  -  Maxwell  used  a  circular  disc  which  vibrated 
torsionally  about  a  perpendicular  axis  through  its  centre.  Two 
similar  fixed  discs  were  placed  one  on  each  side  of  the  vibrating 
disc,  and  the  gas  occupied  the  intervening  space.  The  disc  would 
obviously  oscillate  more  slowly  in  a  viscous  gas  than  in  one  which 
possessed  small  viscosity ;  and  the  quantity  r  may  be  determined 
from  the  results  of  such  experiments.  The  mathematical  investiga- 
tion is  somewhat  more  difficult  than  we  can  venture  to  introduce 
into  an  elementary  work. 

This  property  varies  very  much  from  one  gas  to  another.  In 
hydrogen,  carbonic  acid,  air,  and  oxygen,  it  increases  from  the  first- 
mentioned  to  the  last-mentioned,  being  about  half  as  great  in 
hydrogen  as  in  air. 

It  increases  very  markedly  also  with  rise  of  temperature. 

The  slow  descent  of  clouds,  or  of  fine  suspended  dust,  in  air  is  due 
to  the  viscosity  of  that  gas.  The  weight  of  a  drop  of  water,  which 
causes  its  descent,  is  proportional  to  the  cube  of  its  diameter ;  but 
the  resistance  which  results  from  viscosity  is  proportional  only  to 
the  first  power  of  the  diameter.  Hence,  if  the  diameter  of  a  drop 
be  reduced  to  one -tenth  of  its  original  value,  the  weight  becomes 
one-thousandth  of  what  it  was  before,  while  the  resistance  is  merely 
reduced  to  one-tenth  of  its  previous  amount.  That  is  to  say,  the  re- 
sistance is  relatively  one  hundred  times  more  effective  than  formerly. 

109.  Diffusion. — When  two  gases,  which  are  not  intimately 
mixed,  occupy  a  certain  volume,  each  gradually  diffuses  itself  through- 
out the  whole  volume,  so  as  to  fill  it  just  as  it  would  have  done  had 
the  other  been  absent.  The  only  effect  of  the  presence  of  the  other 
gas  (on  the  presumption  that  there  is  nothing  of  the  nature  of 
chemical  action  between  them)  is  that  the  time  taken  by  the  first  to 
uniformly  fill  the  space  is  greatly  increased.  The  above  process  is 
called  diffusion,  and  the  corresponding  property  is  diffusivity. 

Experiment  shows  that  the  quantity  of  gas  which  passes  in  time 
t  through -an  area  &,  perpendicular  to  which  the  rate  of  variation  of 
density  per  unit  of  length  is  r,  is  proportional  conjointly  to  t,  a,  and  r. 
Hence,  if  q  be  this  quantity,  we  have 

q  =  Srta, 

where  8  is  a  constant  (the  diffusivity,  or  co-efficient  of  inter-diffu- 
sion) the  magnitude  of  which  depends  upon  the  nature  of  the  gases. 
If  r,  t,  and  a  are  each  unity,  we  get 

q  =  t, 

and  so  we  define  the  diffusivity  as  the  quantity  of  tlic  substance 

9—2 


132  A   MANUAL    OF    PHYSICS.     . 

which  passes  per  unit  of  time  through  unit  area  across  which  the 
rate  of  variation  of  density  of  the  substance  per  unit  length  is 
unity.  The  quantity  r  is  generally  called  the  '  concentration- 
gradient,'  and  may  be  written  in  the  form  dpjdx,  where  p  is  the 
density  and  x  is  measured  along  the  line  drawn  in  the  direction  in 
which  the  diffusion  is  taking  place. 

We  shall  find  afterwards  that  the  kinetic  theory  of  gases  leads  to 
the  conclusion  that  the  co-efficient  of  interdiffusion  of  gases  should 
be  approximately  in  inverse  proportion  to  the  geometrical  mean  of 
the  densities  of  the  two  gases  under  one  atmosphere  of  pressure. 
The  figures  in  the  first  column  of  the  accompanying  table  give  relative 
values  of  $  for  pairs  of  gases,  the  relative  values  of  the  reciprocals  of 
the  geometrical  means  of  which  are  given  in  the  second  column : 

Carbonic  Acid  and  Air          ...         ...        1         ...        1. 

Carbonic  Acid  and  Carbonic  Oxide  ...        1         ...        1. 

Carbonic  Acid  and  Hydrogen          ...     3'9         ...     3'8. 

Carbonic  Oxide  and  Hydrogen        ...     4*6         ...     4*8. 

Oxygen  and  Hydrogen         ...         ...     5*2         ...     4*5. 

The  greatest  deviation  occurs  with  oxygen  and  hydrogen.  This  is 
probably  due  to  molecular  action  between  these  gases. 

110.  Effusion. — The  phenomena  presented  in  the  passage  of  gases 
through  the  pores  of  solids  are  of  great  interest,  and  have  been 
elaborately  investigated  by  Graham.  The  simplest  results  are  obtained 
when  the  solid  is  practically  of  .infinite  thinness  and  is  non-porous, 
but  has  a  small  hole  drilled  through  it.  If  a  gas  is  kept  under 
constant  pressure  at  one  side  of  the  solid,  while  a  vacuum  is 
preserved  at  the  other  side,  the  process  of  passage  of  the  gas  is 
called  effusion.  The  theoretical  treatment  of  the  question  is 
extremely  simple.  The  work  done  in  the  transference  of  unit 
volume  of  the  gas  is  (§  62)  numerically  equal  to  p,  the  pressure ; 
for  the  total  pressure  on  unit  area  then  acts  through  unit  distance. 
And  the  kinetic  energy  acquired  is  ^-pv2,  where  p  is  the  density  or 
mass  per  unit  volume,  and  v  is  the  speed  of  the  escaping  gas. 
Hence  the  speed  of  escape,  and  therefore  the  quantity  of  the 
substance  which  passes  through  in  one  unit  of  time,  is  inversely  as 
the  square  root  of  the  density.  The  observed  and  calculated 
quantities  for  four  substances  are  given  in  the  subjoined  table : 

Observed.  Calculated. 

Carbonic  Acid        0'835  ...     0-809. 

Oxygen        0'952  ...     0-951. 

Air 1  ...     1. 

Hydrogen 3'623  ...     3'802. 


PROPERTIES    OF    GASES.  133 

It  appears  that  hydrogen  and  carbonic  acid  pass  through  more 
rapidly  and  more  slowly  respectively  than  the  above  law  would 
indicate.  The  reason  for  this  will  be  seen  in  next  section. 

111.  Transpiration. — When  the  non-porous  septum,  above  referred 
to,  is  not  thin,  the  small  aperture  becomes  a  tube  of  exceedingly 
fine  bore,  and  the  gas  passes  through  by  transpiration.     Graham 
found  that  the  rate  of  passage  was  altogether  independent  of  the 
nature   of  the   substance  forming    the  walls  of    the  tube.     This 
suggests  that  a  layer  of  the  gas  becomes  deposited  upon  the  interior  of 
the  tube,  so  that  the  gas  has  really  to  flow  through  a  tube  composed 
of  its  own  substance  in  a  highly  condensed  state.      [It  is  well  known 
indeed  that  most,  and  probably  all,  solids  have  a  great  power  of 
condensing  gases  on  their  surfaces  or  within  their  pores.]     Hence 
we  would  expect  that  transpiration  is  a  process  which  depends  upon 
the  viscosity  of  the  gas.     This  is  borne  out  by  the  fact  that  the  rates 
of  transpiration  of  oxygen,  air,  carbonic  acid,  and  hydrogen  are, 
in  increasing  magnitude,  in  the  order  in  which  these  gases  are 
here  named,  being  fully  twice  as  great  in  hydrogen  as  in  air — an 
order  which  is  the  exact  reverse  of  their  order  as  regards  viscosity. 
Hence  we  conclude  that  the  abnormality  in  the  rates  of  effusion  of 
hydrogen  and  carbonic  acid  was  due  to  viscosity,  the  hole  in  the  thin 
plate  acting  to  some  extent  as  a  short  tube. 

112.  When  the  pores  of  a  substance  through  which  a  gas  passes 
are  extremely  fine  (as  in  fine  unglazed  earthenware),  the  rate  of 
passage  follows  the  ordinary  law  of  diffusion  or  effusion,  i.e.,  gases 
pass  through  at  rates  which  are  inversely  as  the  square  roots  of  their 
densities.     Hence  we   have   a  means — known   as  the  method  of 
Atmolysis — by  which  to  separate  a  mixture  of  two  gases  of  different 
densities.     If  the  mixture  be  placed  inside  a  porous  earthenware 
vessel,  the  less  dense  gas  passes  through  most  quickly,  so  that,  when 
the  process  has  gone  on  for  some  time,  we  have  two  portions  of  gas, 
one  containing  in  most  part  the  less  dense  gas,  the  other  composed 
mostly  of  the  denser  one.     The  process  may  be  re-applied  so  as  to 
separate  the  two  constituents  to  any  desired  extent. 

It  has  been  already  mentioned  (§  79)  that  carbonic  oxide  passes 
rapidly  through  red-hot  iron ;  and  hydrogen  passes  through  palla- 
dium, and  even  platinum,  at  ordinary  temperatures. 

In  some  cases  the  gas  combines  chemically  with  the  substance  on 
one  side,  diffuses  through  it,  and  is  given  off  on  the  other  side. 
This  occurs  with  india-rubber. 


CHAPTEK  X. 

PROPERTIES     OF     LIQUIDS. 

113.  Compressibility . — Liquids,  like  gases,  convey  sound,  and  are 
therefore  compressible  and  elastic.  But  they  differ  from  gases,  in  that 
their  compressibility  is  usually  extremely  small.  They  differ,  also, 
as  widely  in  respect  of  the  law  of  compression.  An  inspection  of  the 
diagram  of  §  278  will  show  that  a  vapour  such  as  carbonic  acid 
becomes  more  and  more  compressible  as  it  approaches  the  liquefy- 
ing stage,  while,  during  liquefaction,  the  compressibility  is  infinite. 
The  change  is  in  the  opposite  direction  when  the  whole  substance 
has  become  liquid ;  the  compressibility  is  extremely  small,  and 
diminishes  as  the  pressure  increases.  For  example,  the  right-hand 
portion  of  the  isothermal  of  21°'5  is  practically  a  straight  line,  and 
therefore  the  quantity  dvldp  is  constant.  But  the  compressibility 
is  dvjvdp ;  and  v  diminishes  as  the  liquefying  stage  is  approached, 
so  that  the  compressibility  increases.  Similar  reasoning  proves  the 
above  statement  regarding  the  liquid  condition. 

The  earlier  determinations  of  the  compressibility  of  liquids  were 
made  by  means  of  an  apparatus  called  the  piezometer,  and  the 
more  perfect  modern  appliances  all  work  on  the  same  principle. 
This  apparatus  consists  of  a  large  glass  bulb,  having  a  narrow  care- 
fully-graduated stem,  which  is  open  at  the  top.  The  internal 
volumes  of  the  bulb  and  of  the  stem  are  accurately  measured.  The 
liquid,  whose  compressibility  is  to  be  determined,  fills  the  bulb  and 
part  of  the  stem.  A  small  column  of  mercury  suffices  to  separate 
the  liquid  inside  the  bulb  from  water  which  fills  a  strong  glass 
vessel,  inside  of  which  the  bulb  is  placed.  The  outer  vessel  is 
closed,  and  pressure  is  applied  by  screwing  in  a  plug  so  as  to 
diminish  the  internal  volume. 

The  pressure  is  communicated  to  the  liquid  inside  the  bulb,  since 
there  is  a  complete  liquid  connection  through  the  stem ;  and  its 
amount  may  be  measured  by  means  of  the  compression  of  air  con- 


PROPERTIES    OF    LIQUIDS.  135 

tained  in  a  glass  tube,  which  is  closed  at  the  upper  end,  and  is 
also  placed  inside  the  outer  vessel. 

If  the  glass  were  incompressible,  the  compression  of  the  liquid 
would  be  at  once  found  by  means  of  the  extent  to  which  the 
mercury  index  descended  in  the  stem.  If  the  liquid  were  incom- 
pressible, while  the  glass  was  capable  of  compression,  the  index 
would  rise.  If  the  liquid  and  the  glass  were  equally  compressible, 
the  position  of  the  mercury  would  not  alter.  Hence  we  see  that 
this  experiment  really  gives  the  difference  between  the  compressions 
of  the  glass  and  the  liquid,  so  that  we  must  first  know  by  experi- 
ment the  compressibility  of  the  glass  of  which  the  bulb  is  formed. 
This,  as  we  shall  see  in  the  next  chapter,  is  a  comparatively  simple 
matter. 

Water  is  compressed  by  about  one  twenty-thousandth  part  of  its 
bulk  per  atmosphere  of  pressure  added.  Unlike  all  other  liquids 
hitherto  observed,  its  compressibility  diminishes  when  its  tempera- 
ture is  raised,  a  minimum  being  reached  about  63°  C. 

The  compressibility  of  all  liquids  is  lessened  by  increase  of 
pressure. 

114.  Elasticity. — All  liquids,  like  gases,  possess  perfect  elasticity 
of   bulk,  and,  in  common  also  with  gases,   have  no  elasticity  of 
form. 

115.  Viscosity  and  Viscidity. — Viscosity  is  very  much  more 
marked  in  liquids  than  in  gases,  and  varies  greatly  from  one  liquid 
to  another.     The  slowness  of  the  descent  of  fine  mud  in  water  is 
due  to  the  viscosity  of  that  liquid,  and  the  slowness  of  the  fall  of 
fine  rain-drops  is  caused  by  the  viscosity  of  air.     Glycerine  is  one 
example  of  an  extremely  viscous  liquid,  while  sulphuric  ether  has 
little  viscosity  in  comparison  with  it. 

One  extremely  simple  method  of  determining  the  viscosity  of 
a  liquid  consists  in  forcing  it  under  pressure  through  a  cylindrical 
tube  of  very  fine  bore.  The  quantity  which  passes  through  per 
unit  time  is  directly  proportional  to  the  difference  of  pressure  per 
unit  length  of  the  tube  and  to  the  fourth  power  of  the  radius,  and 
is  inversely  as  the  co-efficient  of  viscosity. 

Viscosity  diminishes  rapidly  with  increase  of  temperature. 

Viscidity  is  a  related  property  in  virtue  of  which  a  liquid  can  be 
drawn  out  into  long  threads.  Other  things  being  equal,  a  liquid  is 
viscid  in  proportion  to  its  viscosity ;  but  the  molecular  forces  pro- 
duce another  effect  besides  viscosity,  which  acts  so  as  to  prevent 
viscidity  (§  125). 

116.  Diffusion. — Under  the  diffusion  of  liquids  we  include  the 
diffusion  of  solutions  of  solids. 


136  A   MANUAL    OF   PHYSICS. 

Diffusion  of  liquids  is  a  very  much  slower  process  than  diffusion 
of  gases.  If  a  solution  of  bichromate  of  potassium  be  carefully 
introduced  at  the  foot  of  a  vessel  containing  water,  the  process  of 
interdiifusion  may  go  on  for  months  before  appreciable  uniformity 
is  attained. 

Many  methods  (electrical,  optical,  etc.)  exist,  by  means  of  which 
the  co-efficient  of  interdiffusion  (the  definition  of  which  is  identical 
with  that  given  in  §  110)  may  be  determined. 

In  one  method,  due  to  Graham,  communication  is  established 
between  two  vessels,  each  of  which  contains  a  liquid  capable  of 
diffusing  into  that  contained  by  the  other.  Special  care  is  taken  to 
avoid  the  production  of  currents  whether  in  the  act  of  establishing 
communication  or  because  of  difference  of  density  of  the  liquids. 
The  communication  is  closed  after  a  definite  time,  and  the  extent  to 
which  diffusion  has  gone  on  is  determined.  A  series  of  precisely 
similar  experiments  is  made,  each  experiment  of  the  series  lasting 
for  a  different  interval  of  time,  and  the  diffusivity  is  determined  from 
the  results. 

117.  Osmose. — Dialysis.  —  Diffusion   of  liquids  can   take   place 
through  animal  membrane,  such  as  a  piece  of  bladder.     The  less 
dense  liquid  passes  through  most  quickly.     If  a  vessel  containing  a 
strong  solution  of  sugar  be  closed  tightly  by  means  of  a  membranous 
substance,  and  then  be  immersed  in  a  vessel  of  water,  the  contents 
of  the  inner  vessel  will  rapidly  increase,  and  may  finally  cause  the 
membrane  to  break.     The  process  of    such  transference  is  called 
osmose. 

Liquids  may  be  broadly  divided  into  two  classes  with  reference  to 
the  readiness  with  which  they  pass  through  animal  membranes. 
Crystalloid  substances,  such  as  common  salt,  sugar,  etc.,  pass  easily 
through  when  in  solution ;  but  solutions  of  colloid  substances,  such 
as  glue,  can  scarcely  pass  at  all.  This  is  the  basis  of  the  process  of 
dialysis,  which  is  used  for  the  separation  of  a  mixture  of  colloid  and 
crystalloid  bodies.  The  mixture  is  separated  from  pure  water  by  a 
portion  of  animal  membrane,  through  which  the  substances  pass  in 
very  disproportionate  quantities.  One  or  two  repetitions  of  the 
process  are  sufficient  to  practically  separate  the  two  constituents  of 
the  mixture. 

The  method  is  essentially  analogous  to  the  method  of  atmolysis, 
which  was  described  in  §  112. 

118.  Cohesion. — Cohesion  is  that  property  which,  apart  from  the 
mere  gravitation  of  the  parts  as  a  whole,  results  in  the  clinging 
together  of  portions  of  matter  whether  of  the  same  or  of  unlike 
kinds.     It  may  be  regarded  as  a  result  of  the  so-called  molecular 


PROPERTIES   OF   LIQUIDS. 


137 


forces.  (See,  again,  §  83.)  When  a  body  has  been  pounded  down, 
so  that  its  parts  have  been  separated  beyond  the  range  of  the  mole- 
cular forces,  cohesion  may  be  brought  about  again  by  the  application 
of  pressure  sufficient  to  place  the  molecules  once  more  within  the 
range  of  their  mutual  forces.  In  the  case  of  a  liquid,  it  is  sufficient 
to  merely  place  the  separated  parts  in  contact.  (For  further  treat- 
ment, see  under  Properties  of  Solids.) 

119.  Capillarity. — It  is  a  well-known  law  of  hydrostatics  that  the 
pressure  (§  75)  has  the  same  value  at  all  points  of  a  fluid  which  are 
at  the  same  level,  so  that  we  should  expect  that  the  level  must  be 
the  same  at  all  surfaces  of  a  continuous  fluid  mass  which  are 
exposed  to  the  atmosphere.  Yet,  in  some  cases,  this  is  far  from 
being  the  fact. 

If  a  fine  capillary  tube  be  inserted  in  some  liquids,  the  leve 
is  higher  inside  the  tube  than  it  is  outside ;  while  in  other  liquids 


FIG.  63. 

the  reverse  is  the  case  (Fig.  63).  Thus  water  rises  inside  a  glass 
tube,  while  mercury  descends. 

These  phenomena  (called  capillary  phenomena)  seem  to  be  in 
direct  violation  of  the  above-mentioned  law  of  hydrostatics,  but  in 
reality  they  are  in  strict  accordance  with  it. 

120.  In  proof  of  this  we  observe,  first,  that  the  surface  of  a  liquid 
which  rises  in  a  capillary  is  always  concave  upwards,  while  the 
surface  of  one  which  descends  is  invariably  convex  upwards. 

Next,  we  observe  that,  if  a  surface,  originally  plane,  is  under 
tension  and  is  curved,  there  must  be  more  pressure  on  the  concave 
than  on  the  convex  side.  Otherwise,  the  surface  would  once  more 
become  plane,  because  of  its  tendency  to  shrink.  Hence,  if  we  can 
show  that  there  is  tension  in  the  surface  films  of  liquids,  it  follows 
that  there  is  more  pressure  on  the  concave  side  than  on  the  convex 
side  of  the  curved  surface  of  a  liquid. 


138 


A   MANUAL   OP   PHYSICS. 


But  it  is  well  known  that  the  surfaces  of  liquids  tend  to  become  as 
small  as  possible.  Many  examples  of  this  fact  may  be  brought  for- 
ward. A  soap-bubble  contracts  of  itself  if  the  air  inside  it  be 
in  communication  with  that  outside  ;  and  the  mere  fact  that  the 
soap-bubble  is  naturally  spherical  constitutes  another  proof,  for  the 
sphere  is  the  minimum  surface  which  can  enclose  a  given  volume. 
For  the  same  reason  rain- drops  are  spherical — which  is  proved  by 
the  perfect  circularity  and  definiteness  of  the  rainbow'.  Again,  if 
some  alcohol  be  dropped  on  a  thin  layer  of  ink,  the  surface  of  the 
ink  will  decrease,  while  that  of  the  alcohol  is  increased,  because  of 
the  excess  of  the  surface-tension  of  ink  over  that  of  alcohol. 

Let  us  suppose,  now,  that  the  surface  of  a  liquid  in  a  narrow  tube 
becomes  hollow  upwards.  Just  underneath  the  outside  plane  surface 
there  is  atmospheric  pressure,  while  just  below  the  curved  surface  the 
pressure  is  less.  At  some  point,  p  (Fig.  64),  lower  down  in  the  tube 


FIG.  64. 


FIG.  65. 


the  pressure  (which  is  there  increased  because  of  the  weight  of  the 
liquid)  is  equal  to  that  of  the  atmosphere.  Hence,  in  strict  accord- 
ance with  hydrostatic  laws,  the  liquid  must  rise  until  the  point 
p  is  at  the  level  of  the  surface  of  the  liquid  outside. 

Similar  considerations  explain  the  depression,  in  a  narrow  tube, 
of  a  surface  which  is  convex  upwards. 

It  only  remains  to  explain  why  the  surface  becomes  curved.  We 
shall  assume  that  the  tube  is  of  glass,  that  the  liquid  is  water,  and 
that  the  surrounding  atmosphere  is  air.  The  surface  of  the  liquid 
in  contact  with  the  glass  is  under  tension,  the  amount  of  which 
per  unit  breadth  of  the  film  we  may  denote  by  KTg.  There  is  also 
a  film  of  condensed  gas  on  the  surface  of  the  glass,  the  tension  of 
which  per-  unit  breadth  we  may  similarly  represent  by  «Tr  The 
particles  of  the  liquid  at  the  edge  will  therefore  be  pulled  upwards  or 


PROPERTIES    OF   LIQUIDS.  139 

downwards  according  us  eT^  is  greater  than,  or  less  than,  .T,.  In 
the  case  assumed,  0T^  is  greater  than  ..T^,  and  so  the  water  surface 
becomes  concave  upwards.  The  tension  0TW  of  the  water  surface 
which  is  in  contact  with  the  air — which  formerly  acted  straight 
outwards  from  the  walls  of  the  tube — now  acts  downwards  at  an 
angle  a  with  the  side  of  the  tube  (Fig.  65).  So  we  have  now  a  total 
downward  tension  JSg  +  JTW  cos  a  per  unit  breadth,  and  equilibrium 
will  ensue  when 

.Tj  +  .T.  cos  a=Jfg. 

121.  In  the  case  of  water  this  equation  is  not  satisfied  even  when 
a  vanishes,  for  .T,  is  greater  than  the  sum  of  ,1,  and  .T, ;  and  so 
the  surface  of  water  in  a  narrow  tube  is  hemispherical.  It  is  very 
essential  to  keep  the  surface  of  the  liquid  free  from  impurity,  and 
to  ensure  that  the  surface  of  the  solid  is  chemically  clean.  The 
slightest  trace  of  grease  might  entirely  prevent  the  liquid  from 
rising. 

The  angle  «,  which  is  called  the  angle  of  contact,  may  be  found 
experimentally  by  the  following  method.  Let  AB  (Fig.  66)  be  a 
plane  plate  of  the  glass  (or  other  solid),  and  let  it  be  dipped  into  the 


FIG.  66. 

liquid  CD.  If  the  liquid  rises  and  wets  the  solid,  making  an  acute 
angle  a  with  it,  it  is  evident  that  when  AB  is  inclined  at  the  angle  a 
to  CD,  the  level  of  the  liquid  is  unaltered  at  that  side  of  the  plate 
which  faces  upwards.  We  may  now  find  a  by  direct  measurement. 
The  figure,  if  turned  upside  down,  corresponds  to  the  case  of  a 
liquid  which  descends  in  a  capillary  tube. 

122.  We  can  now  determine  the  height  to  which  a  liquid  will 
rise  in  a  tube  of  given  bore.  Let  r  be  the  radius  of  the  tube,  and 
let  T  be  the  tension  per  unit  breadth  of  the  surface  separating  the 
liquid  from  the  air,  while  a  is  the  angle  of  contact.  T  cos  a  is  the 


140  A   MANUAL   OF   PHYSICS. 

upward  pull  per  unit  breadth,  and  hence  2;rr  T  cos  a  is  the  total 
upward  pull.  This  is  balanced  by  the  weight  of  the  raised  liquid. 
Therefore,  h  being  the  mean  height  to  which  the  liquid  rises  over  the 
outside  level,  while  p  is  the  density  of  the  liquid,  and  g  is  the  value 
of  gravity,  we  have 

2?rr  T  cos  a  =  7rr2hpg. 
This  gives 

h=  2T  cos  a 
pgr 

Hence  the  height  is  inversely  proportional  to  the  radius  of  the  tube. 
When  the  liquid  rises  between  two  parallel  plates  of  breadth  b. 
placed  at  a  distance  d  apart,  the  above  equation  becomes 

26T  cos  a  =  bdhpg. 
So  in  this  case 

h=  2T  cos  a  - 
pgd 

that  is,  while  the  law  is  the  same  as  formerly,  the  height  will  only 
be  the  same  when  the  distance  between  the  plates  is  equal  to  the 
radius  of  the  circular  tube. 

We  can  determine  the  value  of  T  by  either  of  the  above  methods, 
provided  we  know  that  of  «. 

The  height  to  which  the  liquid  rises  is  inversely  proportional  to 
d.  And  since,  if  the  plates  be  not  parallel,  but  be  placed  in  contact 
along  one  of  their  vertical  edges,  d  will  be  proportional  to  the 
distance  from  the  common  edge,  the  liquid,  rising  highest  where  d 
is  smallest,  will  meet  each  plate  in  a  curve  which  is  a  rectangular 
hyperbola.  The  axes  of  the  hyperbolas  will  coincide  respectively 
with  the  common  edge,  and  with  the  lines  in  which  the  level 
surface  of  the  liquid  meets  the  plates. 

123.  The  results  of  §  120  enable  us  to  explain  the  strong  '  attrac- 
tion '  of  two  parallel  glass  plates  between  which  a  drop  of  water  is 
placed.  For,  since  the  water  becomes  concave  outwards,  the 
pressure  inside  it  is  less  than  that  of  the  atmosphere,  and  hence 
the  plates  are  pressed,  not  attracted,  together.  The  lifting  of  a 
stone  by  means  of  a  leather  '  sucker  '  is  similarly  explained. 

The  plates  would  be  apparently  repelled  apart  if  the  liquid  did 
not  wet  them,  i.e.,  if  it  became  convex  outwards.  And  it  is  easy 
to  prove  that  this  would  result  also  if  only  one  plate  were  wet.  The 
liquid  would  rise  to  a  greater  height  outside  the  one  plate,  and 
would  descend  farther  outside  the  other,  than  it  would  rise  or  fall 
in  the  interior  space.  At  the  par^t  ab  (Fig.  67)  there  is  less  than 


PROPERTIES    OF    LIQUIDS. 


141 


atmospheric  pressure  outside  (for  the  liquid  is  concave  upwards), 
while,  inside,  there  is  atmospheric  pressure,  Also,  at  cd,  there  is 
more  pressure  on  the  inner,  than  on  the  outer,  side.  Thus,  from 
both  causes,  the  plates  are  pressed  apart. 

124.  In  the  proof  of  §  73  we  may  suppose  that,  instead  of  a  cord 
stretched  in  a  circular  tube,  we  have  a  film  of  unit  breadth  stretched 
over  a  cylinder.  The  pressure  per  unit  area  will  therefore  be  T/K. 

If  the  film  be  stretched  over  a  spherical  surface  of  the  same 
radius  K,  the  pressure  would  have  the  value  2  T/E,  for  there  are 
now  equal  curvatures  in  two  directions  at  right  angles  to  each 
other.  [The  investigation  of  §  122  furnishes  a  special  proof  of 
this.  For  if  cos  a  —  1,  the  surface  of  the  liquid  is  hemispherical  in 
the  circular  tube,  and  is  cylindrical  in  the  space  between  the 
parallel  plates.  But  the  formulae  show  that,  if  the  radii  of  the 


d 


FIG.  67. 


— 

i 

,n 

h 

"  H 
FIG.  68. 

cylinder  and  the  sphere  are  equal,  the  liquid  rises  twice  as  high  in 
the  tube  as  it  does  between  the  plates.  In  other  words,  the 
pressure  towards  the  centre  of  curvature  of  the  film,  which  supports 
the  weight  of  the  elevated  liquid,  is  twice  as  great  in  the  first  case 
as  in  the  second.] 

In  a  soap-bubble,  we  must  remember  that  the  liquid  film  has 
two  surfaces ;  so  that,  when  the  bubble  is  spherical,  the  air  in  the 
interior  is  subjected  to  a  pressure  which  is  greater  than  that  on  the 
outside  by  the  quantity  4T/E. 

The  above  considerations  indicate  a  method,  due  to  Sir  W, 
Thomson,  of  measuring  the  value  of  T.  A  capillary  tube  is  inserted 
in  the  bottom  qf  a  vessel  B,  (Fig.  68)  which  is  partly  filled  with  the 
given  liquid,  and  is  connected  by  a  syphon  with  a  vessel  A  which 
also  contained  the  given  liquid.  By  raising  or  lowering  A,  the  level 


142  A   MANUAL   OF   PHYSICS.  , 

of  the  liquid  in  B  may  be  altered  at  pleasure.  The  liquid  will  pass 
through  the  capillary  tube,  and  will  gather  into  a  drop  at  the  lower 
end  of  it ;  but  this  drop  will  not  fall  away  unless  the  difference  of 
level,  7t,  between  its  lowest  point  and  the  free  surface  of  the  liquid 
in  B  is  too  great.  The  inward  pressure  per  unit  surface  of  the  drop 
is  p  +  2T/r,  when  p  is  the  atmospheric  pressure  and  r  is  the  radius 
of  the  drop  (measured  by  micrometric  methods).  The  outward 
pressure  per  unit  surface,  due  to  the  weight  of  the  liquid  and  the 
pressure  of  the  atmosphere,  is  j»  +  Us,  where  s  is  the  specific  gravity 
of  the  liquid.  Hence  2T  =  rhs. 

125.  We  have  hitherto  regarded  surface-tension  as  an  observed 
fact  merely,  but  it  is  easy  to  see  that  it  is  a  necessary  result  of  the 
mutual  potential  energy  of  molecules,  or,  as  we  may  put  it,  of  the 
molecular  forces.  Let  p'  (Fig.  69)  be  a  molecule  in  the  liquid, 
situated  at  a  greater  distance  from  its  surface  than  the  range  through 
which  the  molecular  forces  are  sensible.  Draw  a'sphere  from^/  as 


FIG.  69. 

centre  with  the  range  of  the  molecular  forces  as  radius.  There  is 
no  mutual  action  between  _p'and  any  molecule  outside  this  sphere  ; 
and  it  is  equally  attracted  on  all  sides  by  the  molecules  inside  the 
sphere.  But  any  particle  p,  which  is  nearer  the  surface  of  the 
liquid  than  the  given  distance,  is  pulled  inwards  on  the  whole  by 
the  molecular  attraction  of  the  interior  particles.  And  this  inward 
pull  on  the  surface  particles  produces  the  same  effect  as,  and  will 
obviously  be  manifested  as,  a  surface-tension,  tending  to  diminish 
the  external  periphery  of  the  liquid. 

126.  The  tension  of  a  sheet  of  india-rubber  increases  in  propor- 
tion to  the  augmentation  of  the  surface,  but  the  tension  of  a  liquid 
film  remains  absolutely  constant  (at  least  through  extremely  wide 
limits)  when  the  area  of  the  surface  is  altered.  If  left  to  itself,  the 
india-rubber  will  contract  until  the  area  of  its  surface  once  more 
attains  its  original  value ;  but  the  liquid  will  contract  until  its 
surface  becomes  as  small  as  possible. 

Consider  a  film  of  breadth  6,  the  tension  of  which  per  unit 


PROPERTIES   OF   LIQUIDS.  143 

breadth  is  T,  so  that  the  total  tension  is  T6  in  the  direction  of  the 
length  of  the  film,  If  the  length  of  the  film  be  increased  by  the 
amount  I,  the  work  done  in  the  process  is  T6Z  (§  62).  But  this  is 
equal  to  TS,  where  S  is  the  increase  of  surface.  Thus  the  work  is 
directly  proportional  to  the  increase  of  area,  and  we  may  look  upon 
T  as  the  amount  of  work  done  per  unit  increase  of  area  instead  of  a 
tension  per  unit  breadth, 

Taking  this  fact  in  conjunction  with  the  result  of  last  section,  we 
can  now  obtain  an  expression  for  the  exhaustion  of  potential  energy 
of  molecular  separation  (the  work  done  by  the  molecular  forces) 
when  two  separate  masses  of  the  same  liquid  are  placed  in  contact 
over  a  given  area.  Let  S  be  the  area,  so  that  2S  is  the  diminution 
of  surface  of  the  two  masses.  The  work  done  is  2TS. 

Let  T  and  T'  be  the  surface-tensions  of  two  different  liquids,  and 
let  t  be  the  tension  of  the  surface  separating  the  two  when  placed  in 
contact.  The  work  done  by  the  molecular  forces  when  they  come 
into  contact  is  obviously  (T  +  T'  -  t}  S.  The  above  result  is  a 
particular  case  of  this,  for,  when  the  two  liquids  are  identical,  we 
have  t  =  o  and  T  =  T'. 

If  in  any  case  the  work  done  is  greater  than  (T  -f-T')S,  i,e.t  if  t 
be  negative,  the  surface  of  separation  of  the  two  liquids  must 
increase.  This  it  may  do  by  becoming  puckered  ;  and,  the  smaller 
the  scale  of  the  puckering,  the  greater  will  be  the  increase  of 
surface.  Thomson  regards  this  invisible  replication  of  the 
separating  surface  as  the  commencement  of  the  process  of 
diffusion. 

127.  The  surface-tension  of  a  liquid  diminishes  rapidly  with  rise 
of  temperature,  and  it  vanishes  entirely  at  the  critical  temperature 
(§§  24,  278). 

The  saturation  pressure  (§  275)  of  the  vapour  of  a  liquid  depends 
upon  the  temperature.  But,  the  temperature  being  fixed,  it  also 
varies  with  the  curvature  of  the  surface  of  the  liquid,  being  greater 
the  more  convex  outwards  the  surface  is.  Hence  small  drops  of 
water  in  a  cloud  evaporate,  the  vapour  being  deposited  upon  the 
larger  ones. 

From  the  fact  that  the  surface-tension  of  liquids  decreases  with 
rise  of  temperature,  we  might  deduce,  by  the  principle  of  stable 
equilibrium  (§  15),  the  result  that  sudden  extension  of  a  film  will 
produce  a  fall  in  its  temperature.  For,  since  the  system  is  in  stable 
equilibrium,  it  follows  that  extension  of  the  film  will  produce  ar\ 
effect  which  results  in  an  increase  of  the  force  resisting  the  exten- 
sion. It  will,  therefore,  cause  diminution  of  temperature,  This  is 
known  to  be  the  case, 


144  A    MANUAL    OF    PHYSICS.  - 

The  principle  of  dynamical  similarity  shows  at  once  that  the 
square  of  the  fundamental  period  of  vibration  of  a  (weightless) 
liquid  sphere  is  directly  proportional  to  the  density  of  the  liquid  and 
to  the  cube  of  the  radius  of  the  sphere,  and  is  inversely  proportional 
to  the  surface-tension.  It  also  shows  that  the  period  of  funda- 
mental vibration  of  a  (weightless)  soap-bubble  is  independent  of  its 
linear  dimensions. 


CHAPTER  XI. 

PROPERTIES     OF     SOLIDS. 

128.  Compressibility  and  Rigidity. — The  compressibility  of  a  solid 
is  defined  in  precisely  the  same  way  as  that  in  which  we  have 
already  denned  the  compressibility  of  a  liquid  or  gas.  It  is  the 
ratio  of  the  percentage  change  of  volume  to  the  change  of  pressure 
which  produces  it.  The  reciprocal  of  this  quantity  is  called  the 
resistance  to  compression,  and  is  usually  denoted  by  the  letter  Jc. 

The  compressibility  is  most  readily  determined  by  measurement 
of  the  alteration  of  length  of  a  rod  of  the  substance  to  which  known 
hydrostatic  pressure  is  applied.  If  p'  be  the  percentage  alteration 
of  length,  the  percentage  alteration  of  bulk  is  approximately  p  = 
3p'.  For  Z,  6,  and  t,  representing  respectively  the  length,  breadth, 
and  thickness  of  a  rod  of  the  given  substance,  the  new  volume  is 
lbt(l-p')3,  which  approximately  is  lbt(l-8p')  =  lbt(l-p).  In  all 
actual  cases  the  value  of  p'  is  so  small  that  any  power  higher  than 
the  first  may  be  neglected.  [It  must  be  observed  that  we  assume 
the  substance  to  be  isotropic,  i.e.,  its  properties  are  independent  of 
direction.  If  this  were  not  so,  p'  might  have  different  values  in 
different  directions.  This  assumption  will  be  adhered  to  throughout 
the  chapter.] 

The  rigidity  of  a  solid  is  the  measure  of  its  resistance  to  change 
of  shape.  Let  ABCD  (Fig.  70)  be  a  cube  of  a  given  solid,  the  edges  of 


FIG.  70. 

which  are  of  unit  length.     Let  equal  tangential  forces,  of  magnitude 
T  per  unit  area,  be  applied  to  the  opposite  faces  AB  and  CD,  and 

10 


146 


A    MANUAL    OF   PHYSICS. 


let  them  act  in  the  directions  indicated  by  the  arrows.  These 
forces  will  produce  shearing  of  the  cube,  but  they  will  also  produce 
rotation  in  a  direction  opposite  to  that  of  the  hands  of  a  watch.  To 
prevent  this  rotation,  tangential  forces  equal  to  the  former  may  be 
applied  to  the  opposite  faces  AD  and  CB.  The  result  of  the 
application  of  this  set  of  forces  is  that  the  square  section  shown  in 
the  figure  becomes  rhomboidal,  the  angles  at  D  and  B  being  made 
less  than  a  right  angle  by  the  same  amount  that  the  angles  at  A  and 
C  are  made  larger  than  a  right  angle.  If  9  be  the  change  of  angle, 
the  rigidity,  which  is  usually  denoted  by  the  letter  n,  is  given  by  the 
formula 

n  =  T/0. 

129.  Let  us  denote  the  three  pairs  of  parallel  faces  of  a  unit  cube 
by  the  letters  A,  B,  and  C.  Similarly  we  shall  speak  of  the  edges 
joining  the  A  faces  as  the  A  edges,  and  so  on. 

Let  unit  normal  pressure  per  unit  area  be  uniformly  applied  to 
the  A  faces.  This  will  diminish  the  A  edges  by  an  amount  Z,  and 
will  increase  the  B  and  C  edges  by  a  common  amount  I'.  Now  let 
unit  normal  tension  per  unit  area  be  applied  to  the  B  faces.  This 
will  increase  the  B  edges  by  the  amount  Z,  and  diminish  the  C  and 
A  edges  by  the  amount  I',  small  quantities  of  the  second  order  of 
magnitude  being  neglected.  The  result  is  that  the  A  edges  and  the 
B  edges  are  respectively  diminished  and  increased  by  the  amount 
Z+Z',  while  the  C  edges  are  unaltered  in  length.  Hence  there  is  no 
alteration  of  volume. 

Now  this  result  might  have  been  produced  by  the  method  of  last 
section.  If  any  point  in  the  face  DC  (Fig.  71)  of  the  unit  cube  be  slid 
forward  relatively  to  a  point  in  AB  through  the  (very  small)  distance 
s,  the  increase  of  length  of  the  diagonal  DB  is  s  cos  45°  =  s/>v/2. 


A  B 

FIG.  71. 


Similarly  the  decrease  of  length  of  AC  is  «/ ^2.  The  given 
tangential  forces  are  obviously  equivalent  to  a  pressure  parallel  to 
AC  of  magnitude  2Tcos45°  =  TA/2  per  area  ^2,  i.e.,  T  per  unit 
area,  and  to  a  tension  parallel  to  DB  of  the  same  magnitude.  If, 


PROPERTIES    OF   SOLIDS.  147 


now,  we  let  T  =  1,  sj  */2  is  the  alteration  of  length  of  the  diagonal 
which  contains  \/2  units,  and  so  s/2  is  the  percentage  change  of 
length  in  the  direction  of  the  diagonals  due  to  unit  tension  parallel 
to  BD  and  unit  pressure  parallel  to  AC.  Hence,  equating  the  results 
of  the  two  methods,  we  get  s  =  2(Z+  1'}.  But  s  =  9,  the  change  of 
angle  of  the  unit  cube.  Hence 


130.  Unit  pressure  per  unit  area  on  the  A  faces  shortens  the  A 
edges  by  the  amount  Z,  and  increases  the  B  and  C  edges  by  the 
common  amount  I'.  The  quantities  I  and  I'  being  extremely  small, 
if  unit  pressure  be  now  applied  to  the  B  and  C  faces,  all  the  edges 
of  the  cube  will  be  diminished  by  the  amount  I  -  11'  =pr.  Hence 


(2). 


The  quantity  1/Z  (the  reciprocal  of  the  percentage  change  of  length 
of  a  rod  under  unit  tension  or  pressure  per  unit  of  its  transverse 
sectional  area)  is  called  Young's  Modulus.  From  (1)  and  (2)  we 
find 


91m 


This  formula  enables  us  to  determine  the  value  of  either  I,  k,  or  n, 
provided  that  we  know  the  values  of  the  other  two.  The  following 
table  gives  the  value  of  I  for  a  few  substances,  the  unit  of  pressure 
per  square  inch  being  the  weight  of  one  pound  : 

Steel      .........  30(10)  -». 

Iron       .........  39(10)-9. 

Copper  (hard)   ......  56(10)  ~9. 

(annealed)        ...  64(10)-tJ. 

Glass  (average)            ...  141(10)-*. 

131.  The  rigidity  n  is  not  found  in  practice  directly  by  the  process 
which  was  described  in  §  128.  It  may  be  found  by  determining 
the  moment  of  the  couple  which  is  required  to  twist  a  cylindrical 
rod  of  the  substance  through  a  given  angle. 

Consider  a  circular  ring  of  this  rod,  of  radius  r,  of  infmitesimally 
small  breadth  dr,  and  of  thickness  dh  =  dr.  If  the  ring  be  divided 
into  little  parts  by  a  series  of  planes  passing  through  the  axis  OP  of 
the  rod,  and  making  angles  equal  to  dr/r  with  each  other,  each 
little  part  will  be  cubical  in  shape,  and  the  number  of  cubes  will  be 
If,  by  the  given  twist  of  the  rod,  the  upper  side  of  any  one 

10—2 


•148 


A   MANUAL   OF    PHYSICS. 


of  these  little  cubes  is  twisted  forward  relatively  to  the  under  one  by 
the  small  angle  d9,  the  consequent  change  of  angle  of  the  little  cube 
is  rdO/dh.  And,  as  the  same  change  of  angle  would  be  produced  in 


FIG.  72. 

a  unit  cube  by  a  tangential  stress  of  the  same  magnitude  per  unit 
area  as  that  which  acts  upon  the  elementary  cube,  we  have 
(§  128) 

rd0_T 

dh  ~n 

Hence  the  total  tangential  stress  acting  on  the  circular  ring,  which 
is  the  product  of  T  into  the  area  of  the  flat  surface  of  the  ring,  has 
the  value 


-yr 

dh 
The  moment  of  the  force  on  the  ring  is  therefore 


where  9  is  the  twist  per  unit  length  of  the  rod,  so  that  dQ  =  Odh. 
Taking  the  integral  of  this  from  the  axis  outwards,  we  find  for  the 
total  moment  of  the  force  required  to  twist  the  rod  through  an  angle 
9  per  unit  of  length  the  quantity 


r  being  the  radius  of  the  rod. 

The  quantity  7rar4/2Z,  where  I  is  the  length  of  the  rod,  is  called 
the  '  torsional  rigidity  '  of  the  rod.  It  is  not,  of  course,  a  specific 
property. 

[It  is  easy  to  deduce  the  result  (1)  by  elementary  considerations. 
Let  us  imagine  the  rod  to  be  divided  into  similar  little  cubes  as 
above.  The  total  tangential  force  (in  a  plane  perpendicular  to  the 

•^  -  _; 

IT  ft. 


PROPERTIES   OF   SOLIDS.  149 

axis  of  the  rod)  on  one  face  of  any  such  cube,  at  a  distance  r  from 
the  axis,  is  proportional  to  r-  ;  and  therefore  the  moment  of  the 
force  is  proportional  to  r3.  But  the  amount  by  which,  with  a  given 
twist,  the  upper  face  of  the  cube  slides  forward  relatively  to  the 
lower  face  is  proportional  to  r.  Hence  the  total  couple,  c,  required 
to  produce  the  twist  is  proportional  to  r4.  But  it  is  also  proportional 
to  9  so  long  as  Hooke's  Law  holds  ;  and  we  may  therefore  write 


where  n  is  the  rigidity,  provided  that  we  make  a  suitable  definition 
of  the  various  units  involved.] 

Even  a  simpler  experimental  method  consists  in  attaching  to  one 
end  of  the  rod  a  body  whose  moment  of  inertia  about  the  axis  of 
the  rod  is  very  great.  Let  the  rod  be  firmly  clamped  in  a  vertical 
position  by  its  upper  end,  the  body  being  attached  to  its  lower  end. 
The  time  of  oscillation  of  the  whole  system  about  this  axis  depends 
upon  the  value  of  n. 

The  moment  of  the  couple  about  the  axis  is  10,  I  being  the 
moment  of  inertia  of  the  system  (§  70).  Hence,  from  (1), 


(2). 


This  equation  asserts  that  the  angular  acceleration  is  proportional  to 
the  angular  displacement,  and  therefore  the  integral  is  (§  51) 


(3), 
a  and  90  being  constants.     If  we  increase  t  by  the  constant  quantity 


the  value  of  9  is  unaltered.     This  means  that  the  periodic  time  of 
vibration  of  the  system  is 


which  furnishes  a  ready  method  of  determining  n.  (It  is  here 
assumed  that  the  rod  is  of  unit  length.  If  it  be  not  so,  the  equation 
will  still  hold,  provided  that  we  divide  the  observed  value  of  T  by 
the  square  root  of  the  length.) 

132.  The  following  table  gives,  in  the  same  units,  the  values  of 
n  for  the  substances  for  which  I  was  given  in  §  130.  From  these 
numbers  the  values  of  k  are  calculated  by  means  of  (3)  §  130,  and 


150  A   MANUAL   OF   PHYSICS.      ' 

the  observed  values  of  &  are  given  in  the  last  column.  There  is  con- 
siderable discrepancy  between  the  calculated  and  the  observed  values, 
but  this  need  not  produce  surprise,  for  there  are  many  causes  of 
variation  in  the  experimental  results.  The  value  of  r  in  the  expres- 
sion for  the  rigidity  is  usually  small,  so  that  a  large  percentage  error 
may  occur  in  its  measurement  ;  and,  even  if  r  were  uniform  through- 
out the  rod,  which  is  rarely  the  case  —  four  times  this  error  will  be 
produced  in  the  calculated  values  of  n,  since  r  is  involved  to  the 
fourth  power.  Again,  the  substances  may  not  be  really  isotropic  ; 
and  the  special  physical  treatment  —  e.g.,  the  drawing  out  of  a  wire 
—  to  which  an  originally  isotropic  substance  is  subjected  will  fre- 
quently make  it  non-isotropic. 


n 

Steel                    ...       121(10)5  ......        450(10)5  ...  284(10)5. 

Iron  (wrought)  ...       112(10)5  ......        120(10)5  ...  213(10)5. 

Copper          64(10)5  to  71(10)5  ...    122(10)5  to  288(10)5  ...  227(10)5. 

Glass  (average)  ...      28'4(10)5  ......         47(10)5  ...  43(10)5. 

Amagat's  observed  values  of  ~k  for  steel,  copper,  and  glass  are 
respectively  220(10)5,  174(10)5,  and  66(10)5. 

133.  The  flexural  rigidity  of  a  bar  in  a  given  plane  is  measured 
by  the  moment  of  the  couple  which  is  required  to  produce  unit  cur- 
vature in  that  plane.  In  similar  and  equal  bars  of  different  substances 
it  is  directly  proportional  to  Young's  modulus  ;  and,  in  different  bars  of 
the  same  substance  and  of  similar  though  unequal  section,  it  is  pro- 
portional to  the  square  of  the  sectional  area.  However  a  bar  be  bent, 
the  locus  of  the  centres  of  inertia  of  all  the  transverse  slices  is  un- 
altered in  length.  The  length  of  all  other  lines  is  either  increased 
or  diminished.  This  shows  why  Young's  modulus  is  involved. 

Consider  a  rectangular  rod,  of  thickness  d  and  of  breadth  6.  Let 
it  be  bent  uniformly  to  unit  curvature,  and  let  us  suppose  that  b  and 
d  are  small  in  comparison  with  the  radius  of  curvature.  If  we 
further  imagine  the  rod  to  be  composed  of  a  very  large  number  of 
rods,  whose  cross-sections,  are  similar  to  that  of  the  given  rod,  and 
whose  lengths  are  equal  to  the  length  of  the  given  rod,  it  is  easy 
to  see  from  similarity  that  the  couple  which  is  required  to  produce 
the  curvature  must  be  proportional  to  6,  the  breadth  of  the  rod 
measured  perpendicular  to  the  plane  of  bending.  Also  the 
elementary  rods  which  are  further  away  from  the  centre  of  curva- 
ture than  the  central  plane  of  the  given  rod  are  extended  in  pro- 
portion to  their  distance  from  that  plane,  while  those  nearer  to 
the  centre  of  curvature  are  shortened  in  the  same  proportion. 
Hence  we  see  that  the  total  force  must  be  proportional  (so  far  as 


PROPERTIES   OF   SOLIDS.  151 

this  effect  goes)  to  cl  and  to  w  —  Young's  Modulus  ;  and  therefore  the 
moment  of  the  couple  must  be  proportional  to  in  and  to  d2.  But 
the  number  of  little  rods,  of  a  given  size,  is  proportional  to  d  ;  and 
therefore,  finally,  the  moment  c  must  be  proportional  to  &,  to  m, 
and  to  d?  —  say 


where/  is  a  constant.  Thus  we  .see  that  the  flexural  rigidity  of  a 
rectangular  bar  is  proportional  to  its  breadth  and  to  the  cube  of  its 
thickness  in  the  plane  of  'bending. 

134.  Elasticity.  —  All  solids  possess  elasticity,  both  of  form  and  of 
bulk,  to  a  greater  or  less  extent.     Within  limits  (which  vary  greatly 
in  different  substances)  the  elasticity  is  perfect,  i.e.,  the  original 
form  or  volume  is  entirely  regained  ;  but  if  too  great  stress  be  applied, 
the  body  will  either  break  or  become  temporarily  or  permanently 
distorted.     Steel  is  a  good  example  of  the  former  class.     Its  limits 
of  elasticity  are  very  wide  apart,  and  it  breaks  before  much  perma- 
nent distortion  is  apparent.     On  the  other  hand,  lead  can  scarcely 
recover  entirely  from  any  distortion  however  slight.     When  perma- 
nent distortion  occurs,  the  molecules  have  set  themselves  into  new 
permanent  groupings.     When  the  distortion  is   only  temporary, 
they  can  resume  gradually  their  old  positions. 

An  elastic  solid,  if  it  be  kept  distorted  for  a  considerable  time 
and  then  be  released  so  slowly  that  it  does  not  vibrate,  does  not  in 
general  at  once  recover  its  original  form,  but  gradually  creeps  back 
to  it.  If  it  be  consecutively  distorted,  first  in  one  sense  and  then 
in  the  opposite,  it  will  slowly  recover  from  the  second  distortion  for 
a  time,  and  then  will  undo  the  quasi-permanent  part  of  the  first. 

The  limits  of  elasticity  depend  to  a  large  extent  upon  the  physical 
treatment  to  which  the  substance  is  subjected.  The  elasticity  of  a 
wire  is  greatly  diminished  if  the  wire  be  kept  oscillating  for  a  long 
period  of  time.  That  this  is  so  is  shown  by  the  fact  that  its  oscilla- 
tions die  away  much  more  rapidly  in  this  case  when  it  is  left  to 
itself  after  being  set  in  oscillation.  The  elasticity  is  said  to  be 
'  fatigued  '  by  the  process. 

135.  We  may  conveniently  define  elasticity  as  that  property  in 
virtue  of  which  stress  is  required  to  maintain  strain  (§  68).     Within 
the  limits  of  elasticity,  the  necessary  stress  is  proportional  to  the 
strain.     This  is  known  as  Hooke's  Law,  and  is  usually  stated  in 
the  form  '  Distortion  is  proportional  to  the  distorting  force.' 

The  constancy  of  the  quantities  n  and  k  in  equations  (1),  §  129, 
and  (2),  £  130,  depends  upon  the  truth  of  this  law;  for  their  con- 
stancy implies  that,  if  the  unit  of  pressure  be  multiplied  in  any 


152  A   MANUAL   OF   PHYSICS.     ' 

proportion,  the  quantities  on  the  left-hand  side  of  the  equations 
will  also  be  increased  in  the  same  ratio. 

The  constancy  of  the  pitch  of  a  note  which  is  given  out  by  a 
musical  instrument  proves  that  Hooke's  Law  is  obeyed  by  the 
vibrating  substance  within  the  given  limits  of  distortion. 

The  rigidity  and  the  resistance  to  compression  will  not  remain 
unaltered  if  the  distortion  be  too  great ;  but,  between  new  limits, 
the  law  will  again  hold,  n  and  &  having  new  constant  values. 

136.  Viscosity. — Viscosity,  or  internal  friction,  is   apparent  in 
solids  as  well  as  in  fluids.     The  vibrations  of  a  tuning-fork  die  away 
at  a  geater  rate  than  can  be  accounted  for  by  the  impartation  of 
energy  to  the  air  in  the   production  of  sound.     Internal  friction 
transforms  the  original  energy  partly  into  the  form  of  heat  in  the 
material  of  the  instrument.     The  phenomenon  of  fatigue  of  elasticity 
is  also  due  to  this  cause,  the  internal  friction  being  increased  by 
the  process  which  induces  '  fatigue.' 

137.  Cohesion. — Tenacity. — Cohesion  is  in  general  much  more 
powerful  in  solids  than  in  liquids. 

The  parts  of  any  body  are  kept  together  both  by  the  force  of 
cohesion  and  by  mutual 'gravitation.  In  small  bodies,  such  as 
stones,  the  part  played  by  gravitation  is  totally  iiegligable  in  com- 
parison with  that  due  to  cohesion.  But,  in  large  masses,  such  as 
the  earth,  gravitation  has  much  the  larger  effect. 

The  force  of  cohesion  has  generally  been  regarded  as  a  molecular 
attribute  distinct  from  gravitation.  But  Sir  W.  Thomson  has 
pointed  out  that  cohesion  may  be  explained  by  means  of  the  gravi- 
tational law. 

If  a  bar  of  lead  be  cut  into  two  parts,  such  that  the  freshly-cut 
surfaces  accurately  fit  each  other,  the  parts  will  readily  reunite  by 
cohesion  when  the  surfaces  are  brought  sufficiently  close  to  each 
other.  Such  a  phenomenon  as  this  could  not  occur  if  matter  were 
continuous  and  of  uniform  density  throughout.  For  the  range  of 
the  molecular  forces  (§  145)  is  so  small  that  only  an  extremely  small 
amount  of  matter  at  one  surface  of  the  'bar  could  sensibly  attract, 
according  to  the  gravitation  law,  a  given  particle  at  the  opposed 
surface.  But,  in  order  to  get  comparatively  a  very  large  mass  at 
one  surface  within  '  molecular  range  '  of  the  given  particle  at  the 
other,  we  have  only  to  suppose  that,  as  regards  density,  matter 
is  intensely  heterogeneous  on  an  invisibly  small  scale  ;  and  thus  the 
molecular  gravitational  force  might  be  sufficiently  great  to  account 
for  cohesion. 

138.  The  property  of  tenacity,  in  virtue  of  which  there  is  resist- 
ance to  the  drawing  asunder  of  the  parts  of  a  body,  is  obviously 


PROPERTIES    OF   SOLIDS.  158 

cohesion  regarded  from  a  different  point  of  view.  We  may  measure 
the  tenacity  of  a  substance  by  the  tension  which  is  required  to 
rupture  a  rod  or  wire  of  that  substance  whose  cross-sectional  area 
is  unity.  The  following  table  gives  its  values,  for  sudden  rupture, 
in  a  few  substances.  Eupture  will  be  slowly  produced  by  some- 
what smaller  (frequently  considerably  smaller)  tensions.  These 
numbers,  however,  can  only  be  used  for  purposes  of  rough  calcu- 
lation, as  the  property  varies  with  the  physical  treatment  and 
chemical  purity  of  the  substance.  They  represent  the  number  of 
kilogrammes  whose  weight  will  produce  sudden  rupture  of  a  rod 
which  has  a  sectional  area  of  one  square  millimetre. 

Lead 2-4  ...          2. 

Tin  2-9  ...          3-6. 

Gold  ...         ...  28  ...  '11. 

Silver  30  ...  16.     ' 

Copper  41  ...  32. 

Iron  65  ...  50. 

Steel  99  ...  54. 

Oak  7 

Ash  12 

The  numbers  in  the  first  and  second  columns  refer  to  the  drawn 
and  annealed  states  of  the  metals  respectively.  The  tenacity  of 
wood  is  of  course  very  much  smaller  when  the  length  of  the  rod  is 
taken  across  the  grain  than  when  it  is  taken  in  the  direction  of  the 
grain. 


CHAPTEE  XII. 

THE    CONSTITUTION    OF    MATTER. 

139.  EARLY  in  the  history  of  science,  discussion  arose  regarding 
the  possibility  of  the  infinite  divisibility  of  matter.  A  drop  of 
water  may  be  subdivided  into  smaller  drops  of  water  ;  but  this 
process  cannot  be  indefinitely  continued,  for  a  point  is  ultimately 
arrived  at  beyond  which  subdivision  cannot  be  carried  without 
alteration  of  the  chemical  nature  of  the  substance.  But  the 
problem  with  which  we  are  now  concerned  goes  deeper  than  this  : 
we  wish  to  know  whether  or  not  we  would  ultimately  reach  an 
indivisible  part,  or  atom,  of  matter. 

No  answer  can  be  given  yet  to  this  question,  though  various 
hypotheses  have  been  framed  regarding  the  ultimate  constitution, 
or  structure,  of  matter. 

One  of  the  most  famous  of  these  hypotheses  is  known  as  the 
Lucretian  Hypothesis  of  hard  atoms. 

According  to  Lucretius,  hard  atoms  exist ;  and  they  are  indivisible 
because  they  are  infinitely  hard.  The  reason  which  he  gave  for 
their  existence  was  that  there  must  be  a  limit  to  the 'decay  of 
matter — which  he  asserted  to  be  a  more  rapid  process  than  the 
agglomeration,  or  building  up,  of  matter.  If  there  were  no  such 
limit,  all  matter  would,  he  said,  have  disappeared  in  the  course  of 
infinite  past  ages.  His  hypothesis  may  be  true,  but  his  assumption, 
upon  which  it  was  based,  is  not  true ;  for  we  know  that  the 
building  up  of  matter  into  larger  parts  is  a  more  rapid  process  than 
its  disintegration. 

Lucretius  further  asserted  that  there  must  be  void  spaces  between 
the  atoms,  otherwise  motion  of  solids  in  fluids,  such  as  that  of  a 
fish  in  water,  could  not  occur.  Here,  again,  his  reason  is  not  con- 
clusive, although  his  conclusion  is  correct.  Motion  of  solids  in 
fluids  could  occur  even  if  the  hard  atoms  were  closely  packed 
together.  This  we  know  by  the  principle  of  fluid  circulation  in 


THE    CONSTITUTION    OF    MATTER.  155 

re-entrant  paths.     But  the  fact  that  all  matter  is  compressible  shows 
that  there  must  be  intervals  between  the  hard  atoms. 

140.  The  atomic  hypothesis  of  Boscovich  is  a  mere  mathematical 
device,  which  enables  us  to  avoid  the  physical  difficulties  of  the 
problem.     Boscovich  looked    upon    the    atoms    as    mathematical 
points  endowed  with  inertia  and  with  attractive  or  repulsive  forces 
varying'  with  the  distance.     These  forces  are  always  attractive  when 
the  distance  exceeds  a  certain  superior  limit,  and  are  always  re- 
pulsive when  the  distance  falls  short  of  a  certain  lower  limit.     They 
become  infinite  when  the  distance  vanishes,  so  that  no  two  atoms 
can  occupy  the  same  position  at  the  same  time.     This  thepry  may 
be  worked  out  so  as  to  determine  the  properties  of  a  continuous 
medium  so  constituted.     It  has  recently  been  developed  by  Sir  W. 
Thomson. 

141.  The  most  recent  atomic  hypothesis — and  the  one  possessed 
of  the   greatest  interest  at  the   present  day — is  the  vortex-atom 
hypothesis  of  Sir  W.  Thomson.     According  to  Thomson,  matter 
consists  of  the  rotating  parts  of  a  perfect  (§  74),  inert  fluid,  which 
fills  all  space. 

Since  the  fluid  is  perfect,  any  part  of  it,  being  once  set  in  motion, 
will  remain  in  motion  for  ever,  and  will  be  completely  distinguished 
from  all  other  parts.  Thus  the  principle  of  conservation  of  matter 
is  an  essential  part  of  the  hypothesis. 

The  ultimate  parts  or  vortices  are  indivisible,  not  from  being 
perfectly  hard  but,  because  it  is  impossible  to  get  at  them  in  order 
to  divide  them. 

The  properties  of  such  vortices  may  be  illustrated  by  means  of 
smoke-rings,  such  as  those  which  are  occasionally  seen  issuing  from 
the  funnel  of  a  locomotive  when  the  door  of  the  furnace  is  suddenly 
closed,  or  those  which  are  at  times  blown  from  the  mouth  of  a 
cannon.  These  smoke-rings  may  readily  be  produced  by  means  of 
a  box  one  end  of  which  is  flexible  and  the  other  end  of  which  is  per- 
forated by  a  circular  opening  three  or  four  inches  in  diameter.  If  a 
strong  solution  of  ammonia  be  sprinkled  on  the  floor  of  the  box,  and 
if  a  vessel  containing  a  strong  solution  of  hydrochloric  acid  gas  (or, 
preferably,  common  salt  on  which  strong  sulphuric  acid  is  poured) 
be  also  introduced  into  it,  dense  white  fumes  of  ammonium-chloride 
are  formed.  A  slight  blow  on  the  flexible  end  of  the  box  is  sufficient 
to  drive  out  a  portion  of  the  air  from  the  interior.  This  part 
revolves  round  and  round,  in  the  manner  which  is  indicated  in 
section  in  Fig.  73,  and  forms  a  complete  circular  vortex,  the  course 
of  which  may  be  traced  for  some  distance  through  the  surrounding 
still  air.  As  the  ring  advances,  its  speed  diminishes,  and  its 


156  A   MANUAL    OF    PHYSICS.    ' 

diameter  increases;    and  at  last  it  disappears,  its  motion   being 
stopped  by  friction. 

If  a  second  ring  be  projected  after  the  former,  with  slightly 
greater  speed  and  nearly  in  the  same  line,  actual  collision  will  not 
ensue,  but  each  will  move  aside  vibrating  as  an  elastic  ring  would 
after  direct  impact.  If  it  be  projected  exactly  in  the  same  straight 
line  as  the  first,  the  second  one  will  grow  smaller  as  they  approach, 
and,  its  speed  increasing,  will  pass  through  it.  If  the  relative  speed 
of  approach  be  not  too  great,  the  two  rings  will  not  separate,  but 


FIG.  73. 

will  continue  to  revolve  round  each  other  in  the  manner  indicated. 
This  corresponds  to  molecular  or  chemical  combination  of  vortex 
atoms. 

Currents  are  produced  in  the  surrounding  fluid  which  flow  for- 
wards in  the  direction  of  motion  through  the  interior  of  the  ring 
and  pass  back  round  the  outside  of  it.  It  is  the  action  of  these 
currents  which  prevents  actual  contact  between  two  impinging 
rings,  and  which  makes  it  impossible  to  divide  a  ring  formed  in  a 
perfect  fluid. 

The  circular  atom  is  the  simplest  which  can  exist,  but  a  vortex - 
atom  may  be  built  up  of  any  number  of  simple,  or  knotted,  rings, 
linked,  or  locked,  together  in  any  manner  whatsoever. 

The  mathematical  difficulties  which  beset  the  investigation  of 
the  motion,  and  the  mutual  action,  of  vortices  are  so  great  that 
they  have  been  overcome  as  yet  only  to  a  very  slight  extent.  But, 
in  so  far  as  they  have  been  overcome,  the  results  are  not  unfavour- 
able to  the  hypothesis. 

The  great  recommendation  of  the  hypothesis  is  that  it  postulates 
nothing  but  the  inert  vertically  moving  fluid.  All  the  known 
properties  of  matter  are  to  be  deduced  from  that  one  postulate. 
Other  hypotheses  assume  the  existence  of  special  inter-atomic 
forces,  and,  if  one  assumed  set  fails  to  produce  required  results,  a 
new  set  can  be  conjured  up  to  suit  the  case. 

Further  adaptations  of  the  vortex  theory  will  be  given  in  next 
chapter,  and  in  Chap.  XXXIII. 

142.  Standing  in  distinct  contrast  with  these  atomic  hypotheses, 
we  have  the  hypothesis  that  matter  is  continuous  but  intensely 
heterogeneous.  The  atomic  hypotheses  may  be  very  roughly  illus- 


THE  CONSTITUTION  OF  MATTER.  157 

trated  by  a  brick  wall  built  without  mortar.  The  separate  bricks 
represent  the  various  atoms,  and  there  are  gaps  between  them. 
The  hypothesis  which  we  now  consider  is  similarly  illustrated  by.  a 
wall  in  which  the  gaps  are  filled  up  by  cement.  Viewed  from  a 
distance,  the  whole  seems  homogeneous  (as  matter  does  even  when 
we  inspect  it  with  the  most  powerful  microscopes)  but,  if  suffi- 
ciently closely  inspected,  it  is  seen  to  be  heterogeneous. 

Heterogeneity  is  a  necessity  whether  atoms  exist  or  not.  This 
is  indicated  by  many  physical  phenomena,  notably  by  the  disper- 
sion of  light,  which,  if  the  undulatory  theory  be  true,  could  not 
occur  were  matter  homogeneous- and  continuous. 

148.  If  we  assume  the  existence  of  molecules  which  exert  action 
in  all  directions  throughout  a  small  range,  we  can  explain  the 
various  phenomena  of  crystalline  structure.  Let  us  suppose  first 
that  the  molecules  are  in  .stable  equilibrium  under  their  mutual 
action  when  their  centres  are  at  a  definite  distance  apart.  We  can 
obviously  make  a  model  of  such  an  arrangement  by  means  of  equal 
spherical  balls  or  marbles,  a  molecule  being  supposed  to  be  situated 
at  the  centre  of  each. 


FIG.  74. 

We  may  lay  down  a  plane  triangular  arrangement  of  marbles 
(Fig.  74),  and  then  build  up  a  solid  by  placing  a  second  set  of 
marbles  above  the  first,  so  that  each  marble  in  the  second  set  fits 
into  the  hollow  between  three  in  the  first  set,  and  so  on.  In  this 
way  we  form  a  regular  tetrahedron. 

This  tetrahedron  has  six  edges,  and,  if  these  edges  be  pared  off 
by  planes  equally  inclined  to  the  adjacent  faces,  we  ultimately  get 
a  cube. 

The  cube  has  eight  vertices,  and,  if  these  vertices  be  pared  off 
by  planes  equally  inclined  to  the  faces  which  meet  at  these  vertices, 
a  regular  octahedron  is  obtained. 

If  the  twelve  edges  of  the  cube  be  bevelled  by  planes  equally 
inclined  to  the  adjacent  faces,  a  rhombic  dodecahedron  is  ulti- 
mately produced. 

It  is  by  no  means  difficult  to  account  in  this  way  for  all  the 


158  A   MANUAL   OF   PHYSICS. 

various  symmetrical  forms  of  crystals  belonging  to  the  cubic 
system ;  and,  if  we  replace  the  component  spheres  by  ellipsoids  of 
revolution,  we  can  account  for  all  known  crystalline  forms. 

We  are  thus  led  to  believe  that  crystalline  bodies  are  built  up  of 
particles  which  set  themselves,  under  the  action  of  their  mutual 
forces,  in  the  position  of  least  potential  energy,  i.e.,  in  the  position 
of  stable  equilibrium.  The  strongest  proof  of  this  which  we  can. 
have  is  contained  in  the  fact  that  the  lengths  of  the  edges  of  any 
given  natural  crystal  are  expressible  as  multiples  of  small  whole 
numbers. 

144.  The  square  order  of  arrangement  (indicated  in  the  above 
figure)  in  which  any  one  sphere  touches  four  spheres  in  the  layer 
immediately  below,  is  in  no  way  different  from  the  triangular 
arrangement  just  considered;  for,  if  we  remove  an  edge  row  of 
spheres  from  the  triangular  pyramid,  it.  is  at  once  evident  that  the 
particles  are  arranged  in  square  order  in  planes  which  bevel  an 
edge  symmetrically. 

In  both  the  triangular  and  the  square  order  of  arrangement,  any 
one  particle  touches  twelve  others.  In  the  triangular  order,  a 
particle  touches  six  others  in  its  own  layer,  and  three  in  each  of  the 
adjacent  layers.  In  the  square  order  a  particle  touches  four  in  each 
of  these  layers. 

It  is  true  indeed  that  all  the  crystalline  forms  of  the  cubic  system 
may  be  explained  by  two  other  methods  of  arrangement,  but  these 
are  of  no  interest  to  us  physically,  as  they  do  not  correspond  to 
that  position  of  most  stable  equilibrium  which  free,  mutually 
attractive,  particles  would  naturally  assume.  Thus  we  might  build 
up  a  cube  first  of  all  on  the  square  order  of  arrangement — a 


oo 
oo 


FIG.  75. 


particle  in  any  one  layer  resting  on  the  top  of  one  in  the  subjacent 
layer ;  and,  from  this  cubic  form,  we  might  then,  as  above,  produce 
the  other  crystalline  forms.  Again,  we  might  start  with  an  open 
square  arrangement  (indicated  in  Fig.  75)  the  particles  in  any  one 
layer  being  so  far  apart  that  another  particle  which  is  set  in  the 
hollow  between  four  will  just  touch  a  particle  which  is  set  in  the 
corresponding  hollow  on  the  other  side  of  these  four.  In  this  way, 


THE    CONSTITUTION    OF    MATTER. 


159 


also,  all  the  crystalline  forms  of  the  regular  cubic  system  may  be 
produced.  In  the  former  method,  any  particle  touches  four  particles 
in  its  own  layer,  and  one  particle  in  each  of  the  adjacent  layers — 
six  in  all ;  in  the  latter  method,  a  particle  touches  none  in  its  own 
layer,  four  in  each  of  the  layers  immediately  preceding  and  succeed- 
ing its  own,  and  one  in  each  of  the  layers  outside  of  these — ten 
altogether.  In  neither  of  these  cases  is  the  equilibrium  so  stable 
as  in  the  case  which  was  considered  in  last  section. 

145.  Many  other  physical  phenomena,  besides  those  which  are 
exhibited  by  crystalline  bodies,  make  it  a  practical  certainty  that 
matter  is  molecular  in  its  structure ;  and  these  phenomena  also  enable 
us  to  obtain  approximate  estimates  of  the  range  of  the  molecular 
forces,  and  of  the  average  distance  between  contiguous  molecules. 

One  such  class  of  phenomena  is  that  exhibited  by  liquid  films. 
We  have  seen  (§  125)  that  the  existence  of  molecular  forces  would 
account  for  surface-tension.  And  we  have  seen,  also  (§  126),  that 
the  surface-tension  remains  practically  constant  until  the  thickness 
of  the  film  is  very  largely  reduced.  There  are  optical  methods 
(§§  218-221)  by  which  the  thickness  can  be  very  accurately  ascer- 
tained. Keinold  and  Riicker  have  shown  by  these  methods  that  the 
surface-tension  of  a  soap-bubble  begins  to  diminish  when  the  thick- 
ness is  between  96  and  45  micro-millimetres  (millionths  of  a 
millimetre,  one  inch  being  about  equal  to  25'4  millimetres).  It 
diminishes  to  a  minimum  when  the  thickness  is  12  micro-milli- 
metres, and  then  increases  again  to  a  maximum. 

Plateau  had  previously  shown  that  the  tension  is  unaltered  when 
the  thickness  is  reduced  to  118  micro -millimetres,  and  he  concluded 
that  the  range  of  molecular  forces  is  less  than  59  micro-millimetres. 
(Maxwell,  however,  has  given  theoretical  reasons  for  the  belief  that 
the  tension  may  not  alter  until  the  total  thickness  of  the  film  is  equal 
to  the  range  of  the  forces,  which  would  make  Plateau's  superior 
limit  118  micro-millimetres.) 

By  measurements  of  the  height  to  which  liquids  rise,  because  of 
the  so-called  capillary  forces,  between  parallel  glass  plates  which 
were  coated  with  very  thin  wedge-shaped  metallic  films,  Quincke 
was  led  to  the  conclusion  that  the  forces  between  the  glass  and  the 
liquid  became  evident  when  the  thickness  of  the  metallic  film  was 
50  micro-millimetres. 

Wiener  has  found  that  the  phase  of  the  vibrations  of  light  (§  243) 
which  is  reflected  from  a  thin  film  of  silver  deposited  on  mica 
begins  to  alter  when  the  thickness  is  12  micro-millimetres,  and  that 
it  was  possible  to  detect  the  presence  of  a  film  of  silver  not  exceed- 
ing 0'2  micro -millimetre  in  thickness. 


160  A  MANUAL   OP   PHYSICS. 

Estimates  of  the  magnitude  of  the  molecular  range  have  also 
been  based  upon  the  thickness  of  the  films  of  gas  which  are  con- 
densed upon  solids,  but  these  are  open  to  great  objection. 

From  all  these  results  we  may  conclude  that  the  order  of  magni- 
tude of  the  range  of  molecular  forces  is  about  50  micro-millimetres, 
or  one  five -hundred-thousandth  part  of  an  inch. 

146.  It  is  possible,  also,  to  obtain  an  estimate  of  the  coarse- 
grainedness  of  matter;  that  is,  of  the  average  distance  between 
molecules. 

If  (see  §  324)  .two  plates  of  different  metals,  say  copper  and  zinc, 
be  placed  in  contact,  each  becomes  electrified  oppositely,  and  so  they 
attract  each  other.  If  the  plates  have  an  area  of  one  square  cen- 
timetre and  be  at  a  distance  of  one-hundred-thousaiidth  of  a  centi- 
metre apart,  they  will,  when  joined  by  a  metallic  connection,  attract 
each  other  with  a  force  of  two  grammes  weight.  Hence  the  work 
done  in  bringing  them  into  this  position  by  means  of  the  electric 
attraction  alone  would  be  2/100,OOOths  of  a  centimetre-gramme.  If 
we  now  build  up  a  cube  of  such  pieces  of  metal,  alternately  zinc 
and  copper,  the  thickness  of  each  being  l/100,000th  of  a  centi- 
metre, and  the  distance  apart  of  each  pair  beihg  l/100,000th  of  a 
centimetre,  the  work  done  by  the  electric  attraction  is  2  centimetre- 
grammes.  If  this  work  were  spent  in  heating  the  mass  of  metal, 
the  temperature  would  rise  by  less  than  the  sixteen-thousandth  part 
of  a  degree  centigrade.  But  if  the  thickness  of  the  plates  and 
their  distance  apart  were  l/100,000,000th  of  a  centimetre,  the  work 
would  be  sufficient  to  raise  the  temperature  of  the  mass  by  nearly 
62°  C.  And,  if  the  plates  and  the  spaces  between  them  were  made 
yet  four  times  thinner,  the  work  would  produce  more  heat  than  is 
given  out  by  the  molecular  combination  of  zinc  and  copper.  Hence 
the  magnitude  of  the  contact-electrification  of  zinc  and  copper 
must  sensibly  diminish  before  the  substances  are  so  finely  divided 
as  we  have  supposed.  But  this  suggests  that  there  cannot  be  many 
molecules  in  a  thickness  of  the  l/100,000,000th  of  a  centimetre— 
possibly  the  coarse-grainedness  may  even  be  on  a  larger  scale. 

We  have  already  seen  (§  126)  that  the  work  required  to  increase  the 
area  of  a  water  film  by  one  square  centimetre  is  numerically  equal 
to  the  tension  of  the  film  per  linear  centimetre.  But  the  magnitude 
of  the  tension  is  about  16  centigrammes  weight  per  linear  centi- 
metre. Hence,  if  we  increase  the  surface  of  a  water  film  by  n 
square  centimetres,  we  have  to  do  n  times  16  centimetre -centi- 
grammes of  work.  But  a  water  film  cools  when  it  is  stretched  ; 
and  Sir  W.  Thomson  has  shown  that,  if  the  temperature  is  to  be 
kept  constant — which  is  necessary  if  the  tension  is  to  remain 


THE  CONSTITUTION  OF  MATTER.  161 

constant — we  must  supply  heat  to  the  film,  to  an  extent  which 
would  require  the  expenditure  of  about  half  as  much  work  again. 
So,  finally,  if  the  area  of  a  water  film  be  increased,  at  constant 
temperature,  by  n  square  centimetres,  work  must  be  expended  to 
the  extent  of  24ti  centimetre-centigrammes.  If  we  start  with  a  cubic 
centimetre  of  water,  and  increase  its  area  (and  therefore  diminish 
its  breadth)  one-hundred-million-and-one-fold,  we  must  expend 
2,400,000,000  centimetre-centigrammes  of  work.  But  this  work,  if 
spent  in  heating  the  liquid,  would  be  more  than  sufficient  to 
completely  volatilize  it.  Hence,  if  the  thickness  can  be  diminished 
to  this  extent,  the  surface-tension  must  greatly  decrease  in  magni- 
tude. And  so  we  conclude  that  there  cannot  be  many  molecules  of 
water  in  a  thickness  of  l/100,000,000th  of  a  centimetre. 

The  fact  of  the  dispersion  of  light  in  its  passage  through  dense 
transparent  media  proves,  as  has  been  already  stated;  the  hetero- 
geneity of  these  media.  Starting  from  certain  assumptions  regard- 
ing the  constitution  of  such  media,  Cauchy  deduced  results  which 
indicate  that  there  are  only  about  ten  molecules  to  the  wave-length 
of  violet  light  (about  4'10-5cm.)  in  ordinary  glass.  This  cannot  be 
admitted  for  many  reasons.  But  Thomson  has  recently  shown  that 
it  is  possible  to  modify  Cauchy's  theory  in  such  a  way  as  to  widen 
the  limit  which  it  sets. 

The  kinetic  theory  of  gases  gives  (§  153)  another,  and  even  more 
certain,  indication  of  molecular  magnitude. 

From  these  four  courses  of  reasoning,  Sir  W.  Thomson  concludes 
that  there  are  not  more  than  109,  nor  less  than  5(10)6,  molecules  per 
linear  centimetre  in  ordinary  liquids  or  solids. 

Another  method  consists  in  measuring  the  thickness  of  the 
dielectric  layer  separating  an  electrolyte  from  the  electrodes.  This 
thickness  is  presumably  the  distance  between  the  molecules  of  the 
liquid  and  the  contiguous  molecules  of  the  solid  electrode.  The 
method  gives  values  of  the  number  of  molecules  per  linear  centi- 
metre which  vary  from  107  to  5(10)8. 

Again,  if  we  suppose  a  cubic  centimetre  of  any  liquid  to  be 
divided  up  by  three  sets  of  n  planes  parallel  to  the  three  pairs  of  faces 
of  the  cube,  the  surface  of  the  liquid  is  increased  by  6n  square 
centimetres.  The  work  which  is  required  to  produce  this  increase 
of  surface  is  (neglecting  the  equivalent  of  the  heat  required  to  keep 
the  temperature  constant)  6wT  centimetre-grammes,  where  T  is  the 
surface-tension  expressed  in  grammes  weight  per  linear  centimetre. 
[It  is  assumed,  of  course,  that  T  is  practically  constant  during  the 
process.  We  have  seen  that  Plateau  found  it  to  be  constant  in  a 
water  film  down  to  a  thickness  of  118  micro-millimetres.  And 

11 


162  A    MANUAL    OF   PHYSICS. 

Eeinold  and  Kiicker  have  shown  that  the  value  of  the  second 
maximum,  which  is  reached  when  the  thickness  is  12  micro-milli- 
metres, only  differs  from  the  former  value  by  about  0*5  per  cent.] 
If  n  be  the  number  of  molecules  per  linear  centimetre,  the  quantity 
6nT  is  approximately  equal  to  the  work  required  to  break  up  the 
cube  of  the  liquid  into  its  constituent  molecules.  It  is  therefore 
equal  to  the  work-equivalent  of  the  latent  heat  (§§  276,  289)  of  the 
liquid.  If  L  be  this  quantity,  we  have,  therefore,  as  an  approxi- 
mation, n  =  L/6T.  The  liquids,  water,  alcohol,  ether,  chloroform, 
carbon  bisulphide,  turpentine,  petroleum,  and  wood  spirit,  have, 
according  to  this  method,  50,  52,  30,  15,  19,  30,  40,  and  70  millions, 
respectively,  of  particles  per  linear  centimetre.  These  numbers  all 
lie  well  within  the  extreme  limits  given  by  Thomson.  Of  course, 
no  stress  is  to  be  laid  upon  the  relative,  or  even  the  absolute,  values 
of  the  figures;  the  point  of  interest  is  the  close  agreement  as  to 
the  order  of  the  unknown  quantity. 


CHAPTEK  XIII. 

THE    KINETIC    THEORY    OF    MATTER. 

147.  THE  first  glimmerings  of  the  idea  that  the  observed  properties 
of  matter  may  be  due  to  motion. occurred  as  far  back  as  the  time  of 
Democritus  and  Lucretius.  But  the  idea  did  not  develop  into  an 
actual  physical  hypothesis  until  Hooke,  and,  later,  Daniell  Bernoulli 
suggested  that  gaseous  pressure  may  be  due  to  the  impact  of  the 
molecules  of  the  gas  upon  the  sides  of  the  vessel  which  contains  it. 
Somewhat  later,  Le  Sage,  as  we  have  already  seen,  applied  the  same 
principle  to  the  explanation  of  gravitation,  and  various  developments 
were  made  by  Prevost  and  Herapath.  In  1848,  Joule  calculated  the 
speed  which  the  particles  of  a  given  gas  must  have  in  order  to  pro- 
duce a  given  pressure.  But  the  full  mathematical  development  of 
the  Kinetic  Theory  of  Gases  is  due  mainly  to  Clausius  and  Maxwell. 

148.  In  the  kinetic  theory  it  is  supposed  that  the  particles  of  a 
gas  are  darting  about  in  all  directions  with  great  average  speed. 
Some  of  the  particles  may,  for  a  short  lime,  have  very  much  smaller 
speed  than  this  average — may  indeed  be  at  rest  for  a  moment ;  and 
others  may  be  moving  with  much  greater  speed  than  the  average. 
This  average  is  the  square  root  of  the  mean  of  the  squares  of  the 
individual  speeds  of  the  various  molecules,  and  is  called  the  mean 
square  speed. 

Collisions  are  supposed  to  occur  amongst  these  particles.  In  the 
interval  between  any  two  successive  collisions  of  a  particle  there  is 
a  certain  average  distance  which  the  particle  describes,  and  this 
distance  is  called  the  mean  free  path.  The  mean  free  path,  under 
ordinary  conditions,  is  large  in  comparison  with  the  diameters  of 
the  molecules,  which  are  regarded  as  being  smooth  hard  spheres 
with  unit  co-efficient  of  restitution. 

The  collisional  force  between  two  molecules  is  assumed  to  be 
repulsive,  just  as  it  would  be  in  the  case  of  elastic  solids.  But, 
actually,  in.  many,  or  even  most,  of  the  so-called  'collisions,' 
true  contact  may  not  occur.  The  real  forces  may  be  attractive,  and 

11—2 


164  A   MANUAL    OF    PHYSICS. 

two  rapidly-moving  molecules  coming  within  range  of  mutual 
attraction  may  whirl  round  each  other  in  sharply-curved  paths,  as  a 
comet  dashes  round  the  sun,  the  result  being  the  same  as  if  actual 
contact  had  taken  place.  Some  contacts  must  occur  unless  the 
molecules  are  infinitely  small,  which  we  cannot  admit. 

Experiments  made  by  Joule  and  Thomson  on  certain  gases  show 
that,  if  the  density  of  each  be  varied  while  its  total  energy  remains 
constant,  the  temperature  is  somewhat  higher  when  the  density  is 
greater.  Hence,  if  (§  150)  equality  of  temperature  in  two  portions 
of  gas  means  equality  of  average  kinetic  energy  per  molecule,  it 
follows  that  the  potential  energy  is  somewhat  less  in  the  denser 
condition.  But  this  indicates  molecular  attraction  at  the  average 
distance  of  the  molecules  experimented  with. 

149.  Gaseous  Pressure.  —  Boyle's  Law.  —  Let  n  be  the  number  of 
molecules  per  unit  volume  which  are  moving  in  a  given  direction 
with  speed  which  differs  extremely  little  from  a  certain  quantity  u. 
The  number  of  such  molecules  which  pass  per  unit  time  across  unit 
area  drawn  perpendicular  to  the  direction  of  motion  is  nu  ;  and,  if 
m  be  the  mass  of  each  molecule,  the  momentum  which  these 
particles  carry  with  them  is  mnn?.  By  Newton's  Second  Law  of 
Motion,  this  must  be  equal  to  the  pressure  produced  by  such  mole- 
cules on  the  side  of  the  vessel.  Hence,  the  square  of  tlie  speed 
being  involved,  we  see  that,  so  far  as  the  pressure  is  concerned,  we 
may  assume  that  each  particle  is  moving  with  the  mean  square  speed. 

The  total  pressure  per  unit  of  area  in  the  direction  considered, 
which  we  may  assume  to  be  that  of  the  #-axis,  is  therefore  7?zN%2, 
where  N  is  the  total  number  of  particles  per  unit  volume,  and 
u2  is  the  mean  value  of  u2  for  all  the  molecules.  Similarly  the 
pressures  per  unit  area  in  the  direction  of  the  y  and  z  axes  may  be 
written  wN^and  mNw2  respective^'.  But,  in  a  gas  which  is  at 
rest  as  a  whole,  all  these  quantities  are  equal  ;  and  so  we  have  for 
the  pressure  p  the  expression  p  =  £wN(«.2  +  y2  +  w2),  which  we 
may  put  in  the  form 


p  =  imNV2  =  ipV2  ......  (1) 

where  V  is  the  mean  square  speed  independent  of  direction  and 
is  the  density  of  the  gas.     By  means  of  this  result,  Joule  calculated 
the  value  of  V  in  various  gases.     In  hydrogen  it  is  somewhat  over 
6,000  feet  per  second.  _ 

We  have  seen  that  V  is  constant  when  the  temperature  is  steady, 
so  that  this  equation  asserts  that  the  density  of  a  gas  is  directly 
proportional  to  the  pressure—  which  is  Boyle's  Law. 


THE    KINETIC    THEORY    OF   MATTER.  165 

150.  Avogadro's  and  Charles'  Laws. — The  equation  (1)  may  be 
written 

pv  =  ±T (2) 

where  v  is  the  reciprocal  of  p,  i.e.,  it  is  the  volume  of  unit  quantity 
of  the  gas.  If  we  compare  this  with  the  expression  (§  266)  py  =  RT, 
we  see  that  the  mean  square  of  the  speed  of  molecular  motion  is 
proportional  to  the  absolute  temperature  as  measured  by  a  gas 
thermometer  (§§  266,  267)  filled  with  the  particular  gas  under 
consideration. 

It  follows  from  the  principles  of  the  kinetic  theory  that,  in  a 
mixture  of  two  gases  in  equilibrium,  the  average  kinetic  energy  per 
molecule  of  each  gas  is  the  same.  And,  if  we  assume  the  truth  of 
Avogadro's  law  that  there  is  the  same  number  of  molecules  in  unit 
volume  of  each  of  two  gases  at  given  temperature  and  pressure, 
(1)  shows  that  in  two  such  gases  the  average  kinetic  energy  per 
molecule  is  identical.  Conversely,  if  we  assume — as  is  done  in  the 
kinetic  theory — that  two  gases  at  the  same  temperature  have  the 
same  average  kinetic  energy  per  molecule,  we  can  deduce  Avogadro's 
law  as  a  consequence.  But  it  must  be  carefully  observed  that  the 
truth  of  Avogadro's  law  does  not  establish  the  truth  of  this  assump- 
tion ;  it  merely  proves  on  this  theory  that  two  gases  at  the  same 
temperature  and  pressure  have  equal  average  kinetic  energy  per 
molecule.  If,  however,  the  gases  rigidly  obey  both  this  law  and 
Boyle's  Law,  the  truth  of  the  assumption  follows :  for  then,  in  any 
one  gas,  mV'2  remains  constant,  no  matter  how  much  p  may 
vary. 

Equation  (2)  shows  that  any  gas  which  obeys  these  laws  will 
expand  equally  for  equal  increments  of  temperature,  provided  that 
the  temperature  be  measured  by  the  average  kinetic  energy  per 
molecule,  which  is  a  generalisation  of  the  above  assumption.  Andj 
further,  with  this  proviso,  the  equation  also  shows  that  any  two 
such  gases  will  expand  proportionately  as  their  common  temperature 
rises.  These  two  results  constitute  Charles'  Law  (§  265). 

The  deviations  from  Boyle's  and  Charles'  Laws  can  be  explained 
on  the  kinetic  theory  if  we  take  into  account  molecular  action 
between  the  particles  at  greater  than  collisional  distances. 

151.  Diffusion. — Thermal  Conductivity. — Viscosity. — The  ques- 
tion of  gaseous  diffusion  has  been  already  discussed  in  Chap.  VIII. 
The  kinetic  theory  asserts  that  each  particle  is  darting  about  with 
great  speed,  and  is  only  prevented  from  moving  rapidly  away  from 
the  neighbourhood  of  its  position  at  any  given  instant  by  means  of 
the  great  number  of  collisions  to  which  it  is  subjected  by  other 


166  A   MANUAL   OF   PHYSICS. 

particles.  These  collisions  change  the  direction  of  motion  of  the 
molecule  an  immense  number  of  times  per  second,  and  thus  the 
diffusion  of  particles  throughout  a  mass  of  gas  is  a  very  slow 
process.  The  law  o&inter-diffusion  given  by  the  theory  is  identical 
with  that  deduced  from  observation. 

The  diffusing  molecules  carry  their  kinetic  energy  with  them, 
and  there  is  interchange  of  energy  during  collision.  This  trans- 
ference of  energy  constitutes  (Chap.  XXIV.)  the  process  of  thermal 
conduction.  And  this  process  is  slightly  more  rapid  than  the 
transference  of  the  molecules  themselves ;  for,  though  a  molecule 
may  be  turned  back  by  a  collision,  its  energy  is  handed  on  by 
means  of  the  molecule  which  collides  with  it. 

If  two  portions  of  gas  are  moving  relatively  to  each  other,  the 
molecules  of  each  inter-diffuse,  and  consequently  there  is  inter- 
change of  momentum.  But,  on  the  average,  the  momenta  of  the 
particles  of  each  gas  are  in  opposite  directions,  and  so  the  relative 
motion  is  gradually  stopped.  This  explains  gaseous  viscosity.  An 
illustration  may  be  taken  from  two  railway  trains  running  on  parallel 
lines  past  each  other.  If  luggage  wer.e  thrown  constantly  from 
each  train  into  the  other,  the  interchange  of  momentum  resulting 
from  the  impacts  might  soon  reduce  the  trains  to  relative  rest. 

152.  Evaporation,  Dissociation,  etc. — As  a  gas  becomes  more 
and  more  condensed  the  mean  free  path  of  its  molecules  becomes 
smaller  and  smaller,  until, -in  the  liquid  state,  its  magnitude  is 
excessively  small  in  comparison  with  the  magnitude  of  the  mean 
free  path  of  a  gaseous  particle.  Still,  in  the  liquid,  the  molecular 
action  is  precisely  of  the  same  character  as  that  of  a  gas.  •  But,  as 
a  result  of  the  comparative  closeness  of  the  molecules,  the  trans- 
ference of  energy  by  impact  (that  is,  the  conduction  of  heat)  is  a 
much  more  rapid  process  than  the  transference  of  the  molecules 
themselves  ;  and  the  rate  of  diffusion  of  the  molecules  of  a  liquid  is 
very  small  in  comparison  with  the  rate  of  diffusion  of  gaseous 
molecules. 

In  a  liquid  some  of  the  quickly -moving  particles  may  escape 
from  the  attraction  of  neighbouring  molecules  and  become  particles 
of  vapour.  At  the  same  time  some  vapour  particles  may  become 
entangled  in  the  liquid.  When  these  two  processes  take  place  at 
the  same  rate,  the  liquid  and  its  vapour  are  said  to  be  in  equilibrium. 
When  the  former  process  occurs  most  rapidly,  the  liquid  is  said  to 
be  evaporating;  and,  when  the  latter  process  preponderates,  the 
vapour  is  said  to  condense. 

Dissociation  occurs  when  an  impact  is  so  violent  as  to  break  up  a 
compound  molecule  into  its  constituent  parts.  Probably  dissocia- 


THE    KINETIC    THEORY   OF   MATTER.  167 

tion  occurs  in  all  fluids  to  a  slight  extent,  even  when  their  tempera- 
ture is  far  below  the  ordinary  temperature  of  dissociation,  but,  in 
this  case,  recombination  occurs  as  rapidly.  As  the.  temperature 
rises,  the  impacts  become  more  violent  and  dissociation  goes  on  at 
a  greater  rate,  but  not  at  so  great  a  rate  that  recombination  cannot 
occur  as  quickly.  When,  finally,  the  so-called  temperature  of 
dissociation  is  reached,  recombination  is  unable  to  balance  dissocia- 
tion, and  the  substance  is  resolved  into  its  components.  If  the 
temperature  is  again  allowed  to  fall,  recombination  may,  or  may 
not,  occur,  depending  on  the  condition  whether  or  not  energy  will 
be  degraded  (§  11)  in  the  process.  If  more  energy  will  be  degraded 
by  the  occurrence  of  recombination  than  by  its  non-occurrence, 
recombination  will  ensue.  [See  §  280.] 

An  important  result  of  the  kinetic  theory  is  that,  in  a  vessel 
filled  with  a  mixture  of  different  gases,  the  final  distribution  of 
each  gas  under  the  action  of  gravity  is  the  same  as  if  no  other  gas 
had  been  present. 

Another  very  important  result  (which,  together  with  the  former, 
is  due  to  Clerk- Maxwell)  is  that  a  vertical  column  of  gas,  when  in 
equilibrium  under  the  action  of  gravity,  has  the  same  temperature 
throughout,  or,  in  other  words,  gravity  has  no  effect  upon  the 
conditions  of  thermal  equilibrium. 

153.  An  expression  has  been  deduced  from  this  theory  by  which 
we  can  calculate  the  length  of  the  mean  free  path  of  a  molecule  in  an 
ordinary  gas,  such  as  air,  in  terms  of  observed  values  of  the 
viscosity,  and  of  the  material  and  thermal  diffusivities.  According 
to  Maxwell  the  value  of  this  quantity  in  the  case  of  hydrogen  is 
3'8(10)~6  of  an  inch  (roughly  four  millionths).  The  theory  also 
shows  that  the  number  of  particles  per  cubic  inch  of  ordinary  gases 
is  about  3(10)20  (that  is,  300  million  million  millions),  and  that 
the  diameter  of  a  (supposed  hard  and  spherical)  molecule  is  about 
2-3(10) -8  inch. 

The  following  results  are  given  by  Maxwell : 

Hydrogen.       Oxygen.     Carb.  Oxide.    Carb.  Acid. 
Mean  square  speed   at  0°   C. 

expressed  in  feet  per  second      6190     ...     1550     ...     1656     ...     1320 
Mean  free  path   in  thousand- 

millionths  of  an  inch  ...       3860     ...     2240     ...     1930     ...     1510 

Number     of     collisions     per 

millionth  of  a  second          ...     17750     ...     7646     ...     9489     ...     9720 
Diameter  per  inch  x(10)-n    ...      2300     ...     3000     ...     3500     ...     3700. 

The  length  of  the  free  path  increases  as  the  density  of  the  gas 
diminishes.  Tait  and  Dewar  first  found  that,  in  a  good  vacuum,  it 


168  A   MANUAL    OF    PHYSICS.     , 

may  amount  to  several  inches.  The  action  of  Crooke's  radiometer 
depends  upon  the  great  length  of  the  free  path.  [This  instrument 
consists  of  four  vanes  of  mica  mounted  at  the  ends  of  two 
light  rods,  which  are  fastened  by  their  centres  to  a  vertical  axis 
which  is  free  to  rotate.  Each  disc  of  mica  lies  in  the  vertical 
plane  which  passes  through  the  rod  to  which  it  is  attached,  and 
the  two  rods  are  placed  at  right  angles  to  each  other.  Also  each 
disc  is  blackened  on  one  side  and  is  bright  on  the  other,  the 
blackened  sides  being  all  similarly  situated  with  regard  to  rotation 
around  the  vertical  axis.  The  whole  is  mounted  inside  a  glass 
vessel  from  which  the  air  is  largely  extracted.  Eadiant  heat  (or 
light)  falling  upon  the  vanes  is  absorbed  more  freely  by  the  black 
surfaces  than  by  the  bright  surfaces,  and  so  the  former  become 
hotter  than  the  latter.  Hence  the  particles  of  air  which  impinge 
upon  them  are  driven  off  again  more  violently  than  are  the  particles 
which  impinge  upon  the  bright  sides,  and  so,  by  the  third  law  of 
motion,  the  necessary  reaction  results  in  rotation. 

The  exhaustion  of  the  air  is  carried  to  such  an  extent  that  a 
particle  which  is  repelled  from  a  disc  rarely  encounters  another 
particle  before  it  strikes  the  side  of  the  vessel.  If  the  exhaustion 
is  carried  too  far  the  effect  is  diminished,  because  the  number  of 
particles  which  strike  a  disc  in  a  given  time  is  lessened.] 

Gaseous  matter  whose  molecules  have  very  large  free  paths  is 
sometimes  called  '  radiant '  matter. 

154.  The  statement  that  the  diameter  of  one  of  these  hypothetical 
(hard,  smooth,  spherical)  molecules  is  2*3  hundred-millionths  of  an 
inch  is  not  intended  to  imply  that  this  is  the  size  of  an  actual 
material  molecule.  It  merely  asserts  that,  on  the  average,  this  is 
about  the  least  distance  to  which  the  centres  of  two  molecules  can 
approach  during  collision. 

But,  even  if  they  be  smooth  and  spherical,  the  molecules  of 
matter  cannot  be  hard ;"  for,  in  this  case,  the  time  of  describing  the 
mean  free  path  must  vary  inversely  as  the  average  speed  of  the 
particles.  But,  on  the  contrary,  Maxwell's  experiments  on  gaseous 
viscosity  show  that  the  time  is  independent  of  the  speed.  This 
would  be  possible  with  soft  elastic  particles. 

There  are  other,  even  more  cogent,  reasons  why  the  molecules 
cannot  be  smooth,  hard,  and  spherical.  Thus  the  great  complexity 
of  the  spectra  (Chap.  XVII.)  of  many  gases  and  vapours  shows  that 
the  molecule  is  a  very  complex  structure  with  a  great  many  degrees 
of  vibrational  freedom.  But  a  hard,  smooth,  spherical  molecule 
has,  in  effect,  only  three  degrees  of  freedom — all  translational.  And 
again,  the  theory  asserts  that  each  additional  degree  of  freedom 


THE    KINETIC    THEORY    OF    MATTER.  169 

which  the  molecule  possesses  requires  that  the  ratio  of  the  specific 
heat  (£  271)  of  the  gas  at  constant  pressure  to  that  at  constant  volume 
shall  be  made  larger ;  and  the  actually  observed  values  of  this 
ratio  are  far  smaller  than  that  indicated  by  the  theory.  Further,  it 
seems  certain  that,  in  a  gas  whose  molecules  are  small,  elastic,  solid 
•  bodies,  the  energy  of  translation  of  the  molecules  must  gradually 
be  converted  into  energy  of  vibration  of  smaller  and  smaller  period, 
so  that  finally  the  particles  would  come  to  rest. 

155.  But  these  difficulties  with  which  the  theory  is  beset  do  not 
lead  us  to  discard  it.     Its  results,  briefly  indicated  in  part  in  the 
preceding  sections,  place  it  upon  far  too  firm  a  basis;  and  we  seek, 
rather,  to  inquire  more  closely  into  the  truth  or  probability  of  the 
fundamental  assumptions  of  the  theory,  and  into  the  correctness  of 
our  deductions  from  them.      The  difficulty  regarding  the  specific 
heats  seems  to  be  due  largely  to  a  too  rapid  theoretical  generalisa1 
tion.  „  And  the  difficulty  regarding  the  transformation  of  the  transla- 
tional  energy  of    elastic  solids   into  vibrational   energy  does   not, 
Sir   W.  Thomson   remarks,  seem   to    apply   to   the    case   of    fluid 
vortices. 

156.  We  are  not,  however,  to  rest  content  with  a  kinetic  theory  of 
gases  only.     We  wish,  if  possible,  to  recognise  all  the  properties  of 
matter  as  the  results  of  motion  of  the  parts  of  a  medium  in  which 
we  postulate  nothing  but  incompressibility  and  inertia. 

We  can  make,  from  rigid  portions  of  matter,  a  complex  which 
possesses  elasticity.  A  gyrostat  (essentially  a  heavy  fly-wheel,  of 
great  moment  of  inertia,  free  to  rotate  about  its  axis)  mounted  on 
gymbals,  but  not  set  in  rotation,  serves  very  well  as  a  model  of  a 
plastic  body.  It  may  be  turned  about  into  any  position  which  we 
please,  and  has  no  tendency  to  re-assume  its  original  position.  But, 
if  the  fly-wheel  be  set  in  rotation  about  its  axis,  the  system  at  once 
acts  as  if  it  possessed  rigidity  and  elasticity.  If  it  is  sharply  struck, 
so  as  to  suddenly  turn  its  axis  to  a  slight  extent  from  its  original 
direction,  it  will  oscillate  rapidly  about,  and  finally  come  to  rest 
in,  its  first  position. 

Again,  by  means  of  rigid  rods  on  which  revolving  fly-wheels  are 
pivoted,  we  can  construct  a  framework,  which  acts  like  an  elastic 
spring,  and  may  represent  an  elastic  molecule.  By  joining  together 
millions  of  such  arrangements,  it  is  possible  to  construct  a  model  of 
an  elastic  solid  through  which  distortional  waves  will  pass,  and 
which,  under  proper  conditions,  will  exhibit,  with  regard  to  these 
waves,  the  same  phenomena  as  those  which  are  shown  when  light 
passes  through  a  medium  placed  in  a  field  of  great  magnetic 
force. 


170  A   MANUAL   OF   PHYSICS.      ' 

The  vortex  theory  shows  that  all  that  can  be  done  by  such  a  model 
might  be  done  by  a  model  in  which  vortices  in  a  perfect  fluid  take 
the  place  of  the  revolving  fly-wheels.  And  further,  with  such 
vortex  molecules  we  can  construct — what  cannot  be  done  by  the 
rigid  fly-wheel  molecules — a  model  of  a  gas,  whose  molecules  exert 
mutual  force. 

Such  considerations  lead  us  to  believe  that  we  may  ultimately  be 
able  to  explain  apparently  statical  properties  of  matter  in  terms  of 
motion. 


CHAPTEE  XIV. 

SOUND. 

157.  THE  word  sound  is  generally  used  with  reference  to  the 
physiological  effect  which  results  from  excitation  of  the  hearing 
organ.  In  physical  science  the  term  is  applied  to  the  external 
cause  of  this  subjective  impression. 

We  are  accustomed  to  say  that  sound  travels  through  a  given 
medium,  whether  solid,  liquid,  or  gas  ;  and  by  this  we  imply  that  the 
particles  of  the  medium  do  not  move  forward  from  the  source  of  the 
sound  to  the  place  at  which  it  is  heard.  Intermittent  impact  of  such 
particles  would  account  for  the  vibration  of  the  tympanum  which  is 
necessary  to  the  production  of  the  mental  impression  of  sound. 
But  it  is  quite  obvious  that  the  particles  of  a  solid  cannot  move  for- 
ward in  the  manner  indicated,  and  it  would  be  unscientific  to 
assume  that  the  method  of  transference  of  sound  in  a  gas  differs 
totally  from  the  method  of  transference  in  a  solid  body,  especially 
when  we  consider  that  sound,  after  travelling  through  a  solid,  may 
be  communicated  to,  and  travel  through,  a  gaseous  medium,  such.as 
air.  We  do  not,  however,  need  to  rely  upon  any  such  semi-meta- 
physical argument,  for  the  speed  of  sound  in  air  is  so  great  that  the 
forward  motion  of  the  particles — did  it  occur — would  correspond  to  a 
hurricane  of  much  greater  violence  than  any  ever  observed.  In  short, 
we  know  that  the  passage  of  sound  through  air  is  not  accompanied 
by  such  motion — witness  the  well-known  fact  that  sound  is  usually 
best  heard  in  still  air. 

But  the  passage  of  sound  through  air  may  communicate  vibratory 
motion  to  objects  immersed  in  it  —  e.g.,  the  tympanum  —  and  it 
therefore  involves  transference  of  energ}r,  which  implies  motion  of 
matter.  This  motion  can  be  nothing  else  than  vibratory  motion  of 
the  particles  of  the  medium,  the  state  of  motion  being  handed  on 
from  particle  to  particle  in  the  form  of  a  wave  which  may  cause 
vibratory  motion  of  any  solid  object  which  it  reaches,  just  as  a 
wave  sent  along  a  stretched  cord  may  cause  motion  of  any  object 
to  which  the  cord  is  attached. 


172  A   MANUAL   OF   PHYSICS.      • 

The  vibrations  of  the  particles  of  a  medium  may  be  transverse  to 
the  direction  in  which  a  wave  travels,  or  they  may  take  place  in  that 
direction.  A  little  consideration  will  show  that,  in  a  sound-wave, 
it  is  the  latter  which  must  take  place.  Let  us  suppose  that  a 
sound  is  started  by  an  explosion  at  a  certain  point.  We  know  that 
the  sound  travels  outwards  in.  all  directions  from  this  point  as 
centre ;  and  we  know  also  that  the  particles  at  the  point  are  driven 
outwards  by  the  explosion,  so  as  to  produce  a  state  of  great  con- 
densation in  the  immediate  neighbourhood.  But,  the  medium 
being  elastic,  the  compressed  portions  expand,  and  so  cause  com- 
pression of  the  adjacent  parts,  and  thus  the  state  of  compression 
travels  outwards  from  the  centre.  Since  a  state  of  rarefaction  is 
produced  at  the  very  centre  of  the  explosion,  the  particles  in  the 
compressed  part  just  outside  rush  back  so  as  to  fill  the  partial 
vacancy,  and  so  the  state  of  compression  is  succeeded  by  a  state  of 
rarefaction,  which  travels  outwards  at  the  same  rate.  Thus  we  see 
that  sound  consists  in  the  propagation  of  a  condensational-rarefac- 
tional  wave,  the  particles  of  the  medium  vibrating  to  and  fro  in  the 
direction  in  which  the  wave  travels. 

[It  is  easy  to  construct  a  model,  which  will  roughly  illustrate  this 
process,  by  means  of  a  row  of  equidistant  balls  which  are  attached 
by  strings  of  equal  lengths  to  a  horizontal  straight  rod.] 

The  distance,  measured  in  the  direction  of  propagation,  from  any 
particle  to  the  next  similarly -moving  particle,  is  called  the  wave- 
length. It  may  be  measured,  for  example,  between  points  of 
maximum  condensation,  or  between  points  of  maximum  rarefaction. 

The  actual  motion  of  any  particle  is  a  simple  harmonic  motion,  and 
the  terms  used  in  the  discussion  of  such  motion  (§  51) — amplitude, 
phase,  period,  etc. — are  therefore  used  here  with  similar  meanings. 

It  is  customary  to  speak  of  a  single  disturbance  propagated 
through  the  air  as  a  noise.  The  term  sound  is  employed  when  a 
periodic  disturbance,  or  series  of  disturbances,  is  propagated — the 
sound  being  musical,  or  non-musical,  according  as  the  disturbances 
are  of  regular  or  of  irregular  period. 

When  a  musical  sound  is  produced,  it  is  usual  to  speak  of  the  note 
or  tone  which  is  given  out,  but  it  is  better  to  limit  the  application  of 
the  latter  term  to  a  simple  sound  of  one  definite  period  only.  We 
shall  see  subsequently  that  no  sound  usually  given  out  by  a  musical 
instrument  consists  of  a  simple  tone  only.  The  term  note  may 
conveniently  be  applied  to  the  composite  sound  which  is  actually 
given  out. 

All  sounds  differ  from  each  other  in  three  points  only — intensity, 
pitch,  and  quality.  These  will  be  treated  in  detail  afterwards. 


SOUND  173 

158.  We  shall  now  proceed  to  investigate  the  propagation  of 
sound  through  any  gaseous  medium. 

For  the  sake  of  simplicity,  we  shall  suppose  that  a  disturbance  is 
propagated  in  the  form  of  a  plane  wave  ;  that  is  to  say,  we  assume 
that  any  continuous  set  of  particles,  the  motion  of  which  is  in  the 
same  phase,  lie  in  a  plane. 

Let  u  be  the  speed  with  which  the  sound-wave  travels.  The 
actual  speed,  v,  of  any  particle  is  the  resultant  of  this  speed  u  and 
the  speed  due  to  the  simple  harmonic  motion  of  the  particle. 
Hence  an  ideal  plane,  which  is  perpendicular  to  the  direction  in 
which  the  sound  is  going,  and  which  moves  in  that  direction  with 
speed  u,  possesses  the  characteristic  that  the  instantaneous  speed  of 
all  particles  which  cross  it  is  constant.  (The  value  of  this  speed 
depends,  of  course,  upon  the  initial  position  of  the  plane.)  A 
statement  equivalent  to  this  is  that  the  plane  moves  so  as  to  be 
always  in  a  position  of  constant  density. 

If  p  be  this  density,  we  may  write 

pv  =  c,  ..........  (1), 

where  c  is  a  constant  ;  for  this  equation  expresses  the  fact  that  the 
total  momentum  of  the  particles  which  cross  unit  area  of  the  plane 
in  unit  time  is  invariable.  But  any  two  such  planes,  which  remain 
•equidistant,  obviously  contain  between  them  a  constant  amount  of 
matter  which  moves  with  constant  total  momentum.  Hence  c 
is  absolutely  constant,  and  so,  by  the  methods  of  Chap.  IV.,  we  get 

from(1) 


If  the  pressure  per  unit  area  of  the  plane  under  consideration  be  jp, 
the  pressure  per  unit  area  of  another  plane  at  a  distance  dx  from  it  is 
(p  -\-  dp/dx  ,  dx).  Hence  the  difference  of  pressure  per  unit  area 
of  two  such  planes  is  -  dp/dx  .  dx.  But,  by  the  second  law  of 
motion,  this  must  be  equal  to  the  instantaneous  rate  of  increase  of 
the  momentum  of  a  column  of  the  gas,  of  unit  section,  contained  at 
the  given  instant  between  these  planes.  Hence,  the  amount  of 
matter  in  this  little  volume  being  pdx,  we  get  pdv/dt  =  -  dp/dx. 
That  is,  pvdv/dx  =  -  dp/dx,  and  so 


dp~     pv  ' 
From  (2)  and  (3)  we  have  at  once 


d 


(4). 


174  A    MANUAL   OF   PHYSICS. 

In  a  gas  which  obeys  Boyle's  Law  and  Charles'  Law,  the  relation 
p  =  Rfy  holds  —  E  being  a  constant  and  t  being  the  absolute  tempera- 
ture. If  the  temperature  be  constant,  this  gives 


and  (4)  takes  the  form 

i,2  =  P  =  IM  ..........  (6). 

P 

But  the  compressions  and  rarefactions  which  take  place  when 
sound  passes  through  a  gas  take  place  so  rapidly  that  the  equation 
p  =  Tltp  does  not  represent  the  actual  conditions.  Instead  of  it  we 
must  write  (§  302) 


where  y  is  the  ratio  of  the  specific  heat  of  air  at  constant  pressure  to 
its  specific  heat  at  constant  volume  ;  so  that,  instead  of  (5),  we  get 


and  (6)  becomes 

^=r^  =  rR*  ........  (6'). 

159.  The  general  equation  (4)  gives  the  value  of  v  under  all  con- 
ditions of  the  substance,  and  therefore  (6')  gives  the  value  of  v 
under  ordinary  conditions  of  the  gaseous  medium,  provided  only 
that  we  put  the  normal  values  of  p  and  p  in  the  expression  on  the 
right-hand  side  of  that  equation. 

Hence  we  see  that,  in  cases  in  which  (6')  is  sufficiently  nearly 
true,  the  speed  of  sound  is  independent  of  the  pressure  ;  and  that, 
in  all  such  cases,  it  is  proportional  to  the  square  root  of  the 
absolute  temperature. 

Also,  in  different  gases  under  equal  pressure  and  at  the  same 
temperature,  the  speed  is  inversely  proportional  to  the  square  root 
of  the  density  provided  that  (6)  is  true.  The  following  table  shows 
how  nearly  the  theoretical  relative  speeds  correspond  to  the  actual 
relative  speeds  as  observed  in  a  few  of  the  more  common  gases  : 

Observed.  Calculated. 

Air        ...         .........     1-000  ......     1-000. 

Carbonic  Acid  ......     0-786  ......     0-811. 

Oxygen  .........     0-953  ......     0-952. 

Hydrogen       .........    3-810  ......    3'770. 


SOUND.  175 

It  must  be  remembered  that  these  results  are  calculated  on  the 
supposition  that  the  gases  are  perfect. 

[The  following  considerations  enable  us  to  deduce,  in  a  much  less 
strict  manner,  the  general  conclusions  arrived  at  by  means  of  the 
investigation  of  §  158. 

The  speed  of  propagation  of  a  given  state  of  compression  depends 
upon  the  readiness  with  which  the  compressed  gas  recovers  its 
original  condition  ;  and  this,  in  a  perfect  gas,  is  proportional  to  the 
resistance  to  compression,  which,  §  104,  is  measured  by  the  pressure. 
But  the  readiness  of  recovery  depends  also  upon  the  mass  which  has 
to  be  moved  before  the  substance  can  expand,  i.e.,  it  depends  upon 
the  density,  being  greater  the  smaller  the  density  is.  Hence  we  see 
that,  in  a  perfect  gas,  the  speed  of  sound  depends  only  on  the 
temperature  ;  for  (the  pressure  remaining  the  same)  the  density 
diminishes  as  the  temperature  increases,  while,  if  the  temperature 
be  constant,  the  pressure  and  density  vary  proportionately.] 

160.  Equation  (4)  gives,  not  merely  the  speed  of  sound  in  a  gas 
but,  the  speed  of  plane  sound-waves  in  any  substance.  The 
resistance  to  compression,  &,  is  by  definition,  §  104,  the  ratio  of  the 
increase  of  pressure  to  the  percentage  decrease  of  volume  which  it 
produces.  But  the  percentage  decrease  of  volume  is  equal  to  the 
percentage  increase  of  density.  Hence  we  have  &  =  (>  .  dp  /dp,  and  so 
(4)  becomes 


In  the  case  of  a  perfect  gas,  Jc  =  p,  and  (7)  becomes  identical 
with  (6). 

The  speed  of  sound  in  water  is  nearly  four  times  as  great  as  it  is 
in  air.  This  was  found  experimentally  by  means  of  observations 
made  at  the  Lake  of  Geneva,  the  waters  of  which  are  compara- 
tively free  from  currents.  A  bell  was  sounded  under  water,  and  the 
mechanism  which  rang  the  bell  simultaneously  fired  a  gun  placed  at 
the  surface  of  the  water  at  the  same  spot.  A  large  receiver  filled 
with  air  was  placed  below  the  surface  at  a  known  distance  from  the 
bell.  An  observer,  who  listened  at  the  extremity  of  a  tube  con- 
nected with  the  receiver,  knew  when  the  sound  which  travelled 
through  the  water  reached  the  receiver.  He  also  observed  the 
instant  at  which  the  sound  of  the  report  of  the  gun  reached  him 
through  the  air;  and  so,  knowing  by  the  flash  of  the  gun  the 
instant  at  which  both  sounds  were  produced,  he  compared  the 
speeds  in  air  and  water. 

In  solids  the  speed  is  still  greater. 


176  A    MANUAL    OF    PHYSICS. 

The  speed  in  air  may  be  roughly  found  by  observing  the  interval 
of  time  which  elapses  between  the  instant  of  seeing  the  flash  of  a 
distant  gun  and  the  instant  at  which  the  report  is  heard.  .If  the  air 
is  not  still,  two  simultaneous  observations  must  be  made,  in  one  of 
which  the  sound  travels  with  the  wind,  and  in  the  other  of  which  it 
goes  against  the  wind.  The  mean  of  the  two  results  must  be 
taken. 

A  more  accurate  method  of  determining  the  speed  in  gases  will  be 
given  afterwards  (§  172). 

161.  The  loUctness,  or  intensity,  of  a  sound  is  due  to  the  kinetid 
energy  per  unit  volume  of  the  medium,  and  so  it  depends  upon 
the  amplitude  of  vibration  of  the  particles,  being  proportional  to  the 
square  of  that  quantity.  For  the  same  reason  it  depends  also,  other 
things  being  equal,  upon  the  density  of  the  medium,  being  greater 
as  the  density  is  greater,  and  vanishing  when  the  density  becomes 
zero.  Thus  no  sound  is  heard  when  a  bell  is  struck  inside  a 
receiver  from  which  the  air  has  been  exhausted  as  completely  as 
possible. 

[For  the  displacement  of  a  particle  which  executes  a  simple 
harmonic  vibration  is 

•    2ff, 
a  sin  7p£, 

where  T  is  the  periodic  time,  and  a  is  the  amplitude.  The  speed  of 
motion,  which  is  the  time-rate  of  variation  of  the  displacement,  is 
therefore 

27T  27T 


and  the  energy  of  the  particle  is 

27T2 


where  m  is  the  mass.  In  an  interval  of  time  r,  which  is  very  large 
in  comparison  with  T,  the  energy  may  be  regarded  as  constant  and, 
equal  to  its  average  value  during  the  period  T.  But  this  is 


SOUND. 


177 


The  first  term  vanishes  since  T  is  very  large  in  comparison  with  T, 
and  hence  the  kinetic  energy  is  mTr-or/T2  per  particle,  or  p?r2a2/T2  per 
unit  volume. 

The  maximum  value  of  the  kinetic  energy  per  particle  is 
2w7r-«/yT2,  and  so  the  largest  possible  value  of  the  kinetic  energy 
per  unit  volume  is  2p7ra2/T2.  Hence  the  total  energy  per  unit 
volume  of  a  system  of  particles  in  simple  harmonic  vibration  is  (in 
an  interval  of  time  which  is  large  in  comparison  with  the  period  of  a 
complete  vibration)  one-half  kinetic  and  one-half  potential.  This 
statement  will  obviously  still  be  true,  without  the  above  restriction 
regarding  the  interval  of  time,  provided  that  the  number  of  particles 
per  unit  volume  is  very  great,  and  that  the  wave-length  of  the  dis- 
turbance is  very  small.] 

In  still  homogeneous  air  sound  spreads  outwards  from  the  source 
uniformly  in  all  directions  ;  and  therefore  particles,  whose  vibrations 
are  in  a  given  phase,  lie  on  the  surface  of  a  sphere,  the  radius  of 
which  grows  uniformly.  Since  the  energy  of  vibration  is  distributed 
over  the  surface  of  this  growing  sphere,  it  follows  that  the  intensity 
diminishes  proportionately  as  the  surface  increases.  It  is  therefore 
inversely  proportional  to  the  square  of  the  distance  from  the  source. 

Sound  travels  faster  than  usual  when  the  air  through  which  it 
moves  is  blowing  in  the  direction  in  which  the  sound  moves,  for  the 
two  speeds  are  simply  superposed.  Similarly,  when  the  wind  is 
blowing  in  a  direction  opposite  to  that  in  which  the  sound  moves, 


ct  a,'        a,"  3C 

FIG.  76. 

the  speed  is  distinctly  diminished.  And,  In  addition  to  this, 
motion  of  the  medium  affects  the  distance  at  which  the  sound  may 
be  audible.  For,  if  ab  represent  the  (vertical)  front  of  a  plane-wave 
of  sound  travelling  with  the  wind  in  the  direction  ax,  it  is  evident, 
since  the  motion  of  the  upper  strata  of  air  is  less  retarded  by 

12 


178 


A  MANUAL   OF   PHYSICS. 


friction  than  that  of  the  lower  strata,  that  the  wave-front  gradually 
becomes  more  and  more  inclined  to  the  vertical  as  it  moves 
forward.  Successive  new  positions  are  indicated  by  the  lines  a'b', 
etc.  And  the  sound,  instead  of  travelling  straight  forwards,  is  thrown 
down  towards  the  ground  in  the  manner  indicated  by  the  curved 
lines  in  Fig.  76.  When  the  wind  blows  in  a  direction  opposite 
to  the  direction  of  motion  of  the  sound,  the  sound  is  thrown  up 
from  the  earth's  surface  so  as  to  be  inaudible  at  comparatively 
short  distances. 

162.  Reflection  of  Sound. — When  sound  strikes  an  obstacle,  it  is 
reflected  in  such  a  way  that  the  reflected  ray  (the  word  is  used  by 
analogy  from  the  phenomena  of  light,  which,  we  shall  see  subse- 
quently, are  due  also  to  wave-propagation)  and  the  incident  ray. 


make  equal  angles  with,  and  lie  in  a  'plane  passing  through,  the 
normal  to  the  surface.  Thus,  if  ab  (Fig.  77)  represent  a  plane 
surface  from  which  the  ray  ec  is  reflected  in  the  direction  cf,  the 
line  cf  lies  in  a  plane  passing  through  ec  and  the  normal  cd,  and 
the  angles  ecd  and  fed  are  equal. 

This  law  is  identical  with  that  which  obtains  in  the  reflection  of 
light,  so  that  all  the  results  which  are  deduced  in  Chap.  XVI.  regard- 
ing the  reflection  of  light  from  surfaces  will  at  once  apply  to  the 
case  of  reflection  of  sound.  The  law  can  be  deduced  as  a  result 
of  the  fact  that  sound  consists  in  wave-motion,  in  precisely  the  same 
way  as  that  in  which  the  corresponding  law  is  deduced  in  §  186  as 
a  result  of  the  wave-theory  of  light. 

Echoes  are  due  to  the  reflection  of  sound  from  buildings,  rocks, 
trees,  clouds,  etc.  They  occur  even  when  there  is  no  visible  object 
to  account  for  their  existence.  In  this  case  the  reflection  must 
occur  at  the  common  surface  of  two  large  masses  of  air  of  different 
densities,  or  containing  very  different  amounts  of  moisture  per  unit 
volume.  If  the  reflecting  surface  be  curved,  the  reflected  sound  may 
be  conveyed  to  a  focus  so  as  to  be  much  more  distinctly  heard  than 
the  direct  sound. 


SOUND. 


179 


163.  Refraction  of  Sound.  —  Since  the  speed  of  sound  has 
different  values  in  media  of  different  densities,  and  since,  in  any  one 
homogeneous  medium,  the  sound  spreads  out  uniformly  in  all  direc- 
tions from  a  centre  of  disturbance,  it  follows  that  the  direction  in 
which  a  ray  is  travelling  is,  in  general,  suddenly  changed  when  the 
sound  passes  from  one  such  medium  into  another.  If  it  pass 
from  a  less  dense  into  a  more  dense  medium,  the  direction  of 
propagation  is  inclined  to  the  normal  at  a  smaller  angle  after  the 
interface  is  passed  than  before ;  and  the  opposite  statement  holds 
when  the  first  medium  is  denser  than  the  second.  This  phenomenon 
is  known  as  the  refraction  of  sound.  The  law  is  that  the  incident 


and  the  refracted  rays  lie  in  one  plane  with  the  normal  to  the 
refracting  surf  ace,  and  make  with  it  angles  whose  sines  bear  a  con- 
stant ratio  to  each  other.  If  ab  (Fig.  78)  represent  the  intersection 
of  the  common  surface  of  the  two  media  by  the  plane  of  the  paper, 
while  cd  is  the  normal,  and  if  eo  be  the  direction  in  which  the  sound 
impinges  upon  the  surface,  while  of  is  the  direction  which  it  takes 
after  entering  the  second  medium,  the  angles  i  —  eoc  and  r=fod  are 
connected  by  the  relation  sin  i=n  sin  r,  where  /*  is  a  constant. 

These  results  are  identical  with  those  which  are  observed  in  the 
refraction  of  light  .(§  187),  and  the  reasoning  of  §  200  may  be 
applied  directly  to  the  present  case. 

A  lenticular  bag,  filled  with  carbonic  acid  gas,  has  been  found  to 
convey  sound  to  a  focus  in  precisely  the  same  manner  that  a  glass 
lens  conveys  light  to  a  focus. 

164.  Diffraction  of  Sound.— When  sound  enters  a  room  by  an 
aperture  such  as  a  window,  it  is  equally  well  heard  at  all  parts  inside 
the  room.  It  bends  round  so  as  to  penetrate  every  portion,  and  does 
not  cast  a  sharp  '  shadow  '  of  an  obstacle  as  light  does.  This  bending 

12—2 


180  A  MANUAL   OF   PHYSICS. 

of  sound  into  the  region  behind  an  obstacle  is  termed  diffraction. 
We  shall  see  afterwards  that  the  same  phenomenon  also  occurs, 
under  suitable  conditions,  in  the  case  of  light. 

But  it  is  possible,  by  proper  means,  to  produce  sound  shadows. 
The  necessary  condition  is  that  the  aperture  through  which  the 
sound  passes,  or  the  obstacle  by  which  it  is  intercepted,  shall  be 
large  in  comparison  with  the  wave-length  of  the  sound.  Thiis  a 
sound  made  at  one  side  of  a  steep  high  bank  may  be  totally  unheard 
at  the  other.  So,  also,  a  sound  may  be  clearly  heard  through  a 
hollow  between  two  mountains,  while  it  is  inaudible  in  the  region 
behind  either  mountain.  And,  indeed,  by  using  sound  of  sumciently 
short  wave-length,  we  can  produce  comparatively  sharp  sound 
shadows  of  obstacles  which  are  only  a  few  inches  in  diameter. 
(For'  an  explanation  of  the  phenomenon,  see  the  discussion  of  the 
diffraction  of  light,  Chap.  XVIII.) 

165.  Interference  of  Sound. — As   sound   consists  physically  of 
waves  of  condensation  and  rarefaction,  it  follows  that  two  sounds 
may  '  interfere  '  with  each  other,  according  to  the  usual  laws  of  inter- 
ference of  waves.     Thus,  if  two  sounds  are  travelling  in  the  same 
direction  through  a  given  medium,  and  if  the  intensity  and  wave- 
length of  these  sounds  are  identical,  there  will  be  no  resultant  dis- 
turbance of  the  medium,  i.e.,  no  sound  will  be  heard,  provided  that 
the  maximum  condensation  due  to  one  of  the  disturbances  occurs 
simultaneously  with  the  maximum  rarefaction  due  to  the   other. 
If  both  condensations,  and  therefore  both  rarefactions,  take  place 
together,  a  sound  of  four  times  the  intensity  will  be  heard,  for  the 
resultant  amplitude  of  vibration  of  the  particles  of  the  medium  is 
doubled.     And,  if  the  wave-lengths  of  the  separate  disturbances  are 
not  precisely  identical,  the  resultant  sound  will  periodically  vary  in 
intensity  from  the  former  (zero)  to  the  latter  (quadruple)  value. 

166.  Pitch. — The  resultant  vibration   of  a   particle  of  air  is,  in 
general,  extremely  complex ;  but,  when  a  pure  tone  is  transmitted 
through   air,  the   vibration   of   each   particle   is  simple  harmonic. 
There   can,   therefore,   be  no    difference    between    one   tone   and 
another  except   such  as  is  due  to  a   difference  in  the   amplitude 
of  vibration  or  to  a  difference  in  the  period  of  vibration.     As  the 
amplitude  of  vibration  determines  the  intensity  of  the  sound,  we 
infer  that  the  pitch  of  a  note  depends  upon  the  frequency  of  vibra- 
tion, i.e.,  upon  the  number  of  vibrations  which  are  executed  per 
second. 

That  this  is  actually  the  case  may  be  roughly  ascertained  by 
means  of  very  simple  apparatus.  If  a  piece  of  cardboard  be 
pressed  against  t*he  edge  of  a  toothed  wheel,  which  revolves  at  a 


SOUND.  181 

definite  rate,  a  sound  is  emitted  which  is  of  a  fairly  definite  pitch. 
As  the  speed  of  the  wheel  increases,  i.e.,  as  the  number  of  impacts 
per  second  between  the  teeth  of  the  wheel  and  the  cardboard 
increases,  the  pitch  of  the  note  rises ;  and,  to  a  note  of  given  pitch, 
there  corresponds  a  definite  rate  of  rotation  of  the  wheel,  and 
consequently  a  definite  rate  of  vibration  of  the  cardboard. 

A  much  more  accurate  proof  is  obtained  by  means  of  the  syren. 
This  instrument  consists  essentially  of  a  perforated  disc  of  metal, 
the  perforations  being  arranged  in  a  circle  round  the  centre,  as  in 
the  figure.  A  tube,  through  which  air  is  driven,  is  placed  behind 
the  disc ;  and,  as  the  disc  revolves,  each  successive  opening  in  it 


FIG.  79. 

comes  opposite  the  end  of  the  tube.  As  each  blast  of  air  passes 
through,  a  state  of  condensation  is  produced,  which  is  succeeded 
by  a  rarefaction  in  the  interval  between  twTo  blasts.  Hence  a 
sound-wave  of  constant  period  is  set  up  when  the  disc  revolves  at  a 
uniform  rate.  The  rate  of  rotation,  and  the  number  of  perforations 
in  the  complete  circle,  being  known,  we  get  at  once  the  number  of 
vibrations  per  second  corresponding  to  any  note  of  given  pitch. 

When  the  disc  revolves  very  slowly,  no  musical  sound  is  heard, 
but  each  separate  air-blast  can  be  distinctly  heard.  As  the  speed 
increases,  the  ear  ceases  to  distinguish  the  separate  pulses,  and  a 
note  of  very  low  pitch  becomes  audible.  The  speed  still 
increasing,  the  pitch  of  the  note  produced  becomes  higher  and 
higher,  and  at  last  it  becomes  so  high  that  the  note  is  no  longer 
audible.  The  limits  of  audibility  vary  considerably  in  different 
observers,  but,  roughly  speaking,  a  sound  becomes  inaudible  as  a 
note  when  the  rate  of  vibration  falls  short  of  20  times  or  exceeds 
70,000  times  per  second.  The  fact  that  a  melody  is  perfectly  heard 
at  different  distances  from  the  source  shows  that  the  speed  of  sound 
does  not  depend  upon  the  wave-length. 
/ "The  apparent  pitch  of  a  note  depends  tipon  the  relative  motion  of 

I ' 


182  A   MANUAL   OF   PHYSICS.     , 

the  hearer  and  the  instrument  upon  which  the  note  is  sounded.  If 
the  hearer  approaches  the  instrument,  the  pitch  of  the  note  seems 
to  be  heightened,  because  more  than  the  normal  number  of  vibra- 
tions reach  his  ear  in  a  given  time  ;  and,  conversely,  if  he  recede 
from  the  instrument,  the  pitch  of  the  note  will  appear  to  be 
lowered.  As  a  particular  case,  if  the  rate  of  recession  were 
greater  than  the  speed  of  sound,  no  sound  could  be  heard. 

167.  Musical  Intervals. — If  any  particular  note  be  taken  as  a 
keynote,  it  is  found  that  there  are  certain  other  notes,  definitely 
related  to  it  as  regards  pitch,  which  produce  specially  pleasing 
effects  upon  the  ear  when  sounded  with  each  other  or  with  the  key- 
note. These  are,  therefore,  the  notes  which  are  employed  in  the 
ordinary  major  and  minor  scales,  and  the  difference  in  pitch  of  any 
two  notes  is  called  the  interval  between  them.  When  two  notes 
have  the  same  pitch,  they  are  said  to  be  in  unison. 

The  chief  interval  is  the  octave,  and  it  is  found  that,  in  order  to 
produce  the  octave  of  any  given  note,  the  rate  of  vibration  must  be 
exactly  doubled.  The  intervals  into  which  the  octave  is  subdivided 
are  not  equal.  The  following  table  indicates  their  values.  The 
first  column  gives  the  names  of  the  intervals  which  separate  each 
note  from  the  keynote,  and  the  relative  rates  of  vibration  in  the 
two  notes  forming  each  interval  are  given  in  the  second  column. 
For  example,  nine  vibrations  are  performed  in  the  higher  of  two 
notes  separated  by  the  interval  of  a  second  in  the  time  in  which 
eight  vibrations  are  performed  in  the  lower : 

Unison       ]-. 

Second       g. 

Minor  Third          ...         ...         ...  £. 

Major  Third          |. 

Fourth        \. 

Fifth  for2(f). 

Minor  Sixth          f  or  a(£). 

Major  Sixth  f  or  a($). 

Minor  Seventh      ...         ...         ...  ^  or  2(|). 

Major  Seventh      */. 

Octave        f. 

A  very  little  consideration  will  show  that  in  order  to  find  the  sum 
of  two  intervals  we  must  multiply  together  the  fractions  given 
above  for  each  interval.  Thus  a  fifth  is  the  sum  of  a  minor  third 
and  a  major  third.  Also,  to  find  the  difference  between  two 
intervals  we  must  divide  the  fraction  corresponding  to  the  larger  by 


SOUND.  188 

the  fraction  corresponding  to  the  smaller.  The  second  method  of 
writing  the  fractions  corresponding  to  the  minor  seventh,  the  major 
and  minor  sixths,  and  the  fifth,  shows  that  the  interval  between  the 
notes  so  indicated  and  the  octave  are  respectively  a  second,  a  minor 
third,  a  major  third,  and  a  fourth. 

168.  Vibrations  of  Bods. — We  shall  assume  that  the  extent  of 
the  vibrations  is  such  that  Hooke's  Law  (§  135)  is  followed.  In  this 
case,  since  the  period  of  vibration  is  independent  of  the  extent  of 
vibration,  a  musical  note  of  constant  pitch  will  be  heard,  provided 
that  the  rate  of  vibration  is  sufficiently  rapid.  Two  kinds  of  vibra- 
tions have  to  be  considered. 

Transverse  Vibrations. — We  see  from  §§  51,  63  that  the  time  of 
simple  harmonic  vibration  of  any  material  system  varies  directly  as 
the  square  root  of  the  mass  to  be  moved  and  inversely  as  the  square 
root  of  the  stress  called  into  play  by  a  given  displacement.  It 
therefore,  in  the  case  under  consideration,  varies  inversely  as  the 
square  root  of  the  flexural  rigidity.  Now  both  the  flexural  rigidity 
and  the  mass  of  a  rod  are  (§  133)  proportional  to  its  breadth,  and  so 
the  period  of  transverse  vibration  is  independent  of  the  breadth  of 
the  rod.  Again,  the  mass  of  the  rod  is  proportional  to  the  length, 
while  the  rigidity  varies  inversely  as  the  cube  of  the  length.  Hence 
the  period  is  proportional  to  the  square  of  the  length.  Similarly, 
it  varies  inversely  as  the  thickness,  since  the  rigidity  is  directly  pro- 
portional to  the  cube  of  the  thickness. 

Hence  we  conclude  that  the  time  of  transverse  vibration  of 
similar  rectangular  rods  is  in  direct  proportion  to  their  linear  dimen- 
sions. And  considerations  of  a  like  kind  enable  us  at  once  to 
extend  this  statement  to  the  case  of  similar  rods  of  any  form  ; 
for  the  masses  of  such  rods  are  in  proportion  to  the  cubes  of  their 
linear  dimensions,  while  their  rigidities  vary  as  the  first  power 
of  the  dimensions. 

Longitudinal  Vibrations.  —  If  a  rod  be  fixed  at  one  end  and 
receive  a  smart  blow  on  the  other  end  in  a  direction  parallel  to 
its  length,  a  wave  of  compression  will  travel  along  it  to  the  fixed  end, 
from  which,  by  reflection,  it  will  return  to  the  free  end.  The  rod 
will  now  extend,  because  of  its  elasticity,  to  a  length  greater  than  its 
normal  length,  and  a  wave  of  rarefaction  will  travel  along  it  to 
the  fixed  end  at  which  it  too  will  suffer  reflection.  The  free  end 
is  obviously  a  loop,  or  place  of  maximum  motion,  while  the  fixed 
end  is  a  node.  Hence  the  length  of  a  wave  is  four  times  the  length 
of  the  rod,  and  the  time  occupied  by  the  disturbance  in  travelling 
from  end  to  end  of  the  rod  is  therefore  one-quarter  of  the  periodic 
time  of  longitudinal  vibration. 


184 


A   MANUAL    OF   PHYSICS.  . 


If  the  rod  be  fixed  at  both  ends,  the  wave-length  is  equal  to  twice 
the  length  of  the  rod,  since  each  end  is  now  a  node. 

In  each  case  the  periodic  time  is  obtained  by  dividing  the  length 
of  the  wave  by  the  speed  of  the  disturbance.  The  speed  is  given  by 
equation  (7),  §  160,  ~k  being  in  this  case  the  constant  called 
Young's  modulus.  From  this  we  see  that  the  period  is  proportional 
to  the  length  of  the  rod  and  the  square  root  of  the  density,  and  is 
inversely  proportional  to  the  value  of  Young's  modulus,  while  it  is 
independent  of  the  sectional  area. 

The  rod  which  is  fixed  at  both  ends  has  twice  the  rate  of  vibra- 
tion of  a  similar  rod  fixed  at  one  end  only.  A  rod  which  is  free  at 
both  ends,  and  has  its  node  at  the  centre,  obviously  vibrates  at  the 
same  rate  as  does  a  like  rod  with  both  ends  fixed. 

169..  A  rod  which  is  fixed  at  one  or  both  ends  may  have  more 
than  one  mode  of  transverse  or  longitudinal  vibration.  In  Fig.  80 
a  represents  the  fundamental  mode  of  vibration  of  a  rod  fixed 
at  one  end,  while  b  and  c  represent  the  modes  which  stand  next  to 
it  in  order  of  simplicity.  This  is  the  case  of  a  vibrating  tuning-fork. 
When  the  fork  is  feebly  excited,  the  fundamental  mode  of  vibration 
is  the  chief  one  which  occurs ;  but,  under  violent  excitation,  the 
forms  6,  c,  and  other  higher  forms,  are  superposed  upon  it.  The 
extra  node  in  b  is  distant  from  the  fixed  end  two -thirds  of  the  whole 


\/ 


i 


c  d  e 

FIG.  80. 


length  of  the  rod,  since  the  free  part  and  the  looped  part  must  have 
a  common  period  of  vibration.  The  length  of  the  free  part  in  b  is 
therefore  one-third  of  the  length  of  the  free  part  in  a,  and  conse- 
quently the  period  of  vibration  in  the  case  b  is  one -ninth  of  the 
fundamental  period.  A  reference  to  the  table  of  §  167  will  therefore 
show  that  the  note  given  out  by  a  fork  vibrating  in  the  mode  b 
differs  in  pitch  from  the  fundamental  note  by  the  interval  of  three 
octaves  and  a  second. 

In  the  figure,  d,  e,  and  /,  show  the  three  simplest  modes  of  trans- 
verse vibration  of  a  rod  which  is  fixed  at  both  ends.  The  interval 
between  the  notes  given  out  by  similar  rods  vibrating  as  in  d  and  e 


SOUND. 


185 


is  two  octaves,  while  the  interval  in  the  cases  indicated  by  d  and  f  is 
the  same  as  between  those  indicated  by  a  and  b. 

The  modes  of  longitudinal  vibration  of  rods  are  precisely. 
identical  with  those  of  air  -  columns,  which  will  be  discussed 
shortly. 

170.  Vibration  of  Plates.  —  A  plate  may  be  set  in  vibration  by 
drawing  a  bow  across  its  edge.     The  modes  of  vibration  depend  on 
the  form  of  the  plate,  and  may  be  greatly  varied  by  forcing  certain 
points  to  lie  upon  nodal  lines  —  which  may  be  done  by  keeping  the 
fingers  in  contact  with  the  desired  points. 

As  in  §  168,  we  can  without  difficulty  deduce  the  result  that  the 
periods  of  vibration  of  similar  plates  (similar,  of  course,  as  regards 
thickness  as  well  as  form)  of  the  same  material  are  in  proportion  to 
their  linear  dimensions.  Also,  as  in  the  section  referred  to,  we  may 
show  that  the  period  is  inversely  as  the  thickness  of  a  plate  of  given 
form  and  area.  By  combining  these  two  results  we  see  that  the 
periods  of  vibration  of  similar  plates  of  the  same  thickness  are  pro- 
portional to  the  squares  of  their  linear  dimension,  that  is,  to  their 
areas. 

171.  Vibrations  of  Strings.  —  The  speed,  v,  with  which  a  distur- 
bance runs  along  a  stretched  cord  is  (^  73)  proportional  to  the  square 
root  of  the  tension,  T,  and  is  inversely  proportional  to  the  square 
root  of  p,  the  mass  per  unit  length  of  the  string.     Hence,  if  X  be  the 
wave-length,  the  time  of  a  complete  vibration  is 


where  p  is  the  density  of  the  material  of  the  string  and  s  is  its  area 
of  cross-section. 

The  figure  below  shows  the  three  simplest  forms  of  vibration  of  a 
stretched  string  which  is  fixecl  at  its  ends.     Any  disturbance  of  the 


FIG.  81. 

string  is  propagated  with  equal  speed  in  both  directions  along  the 
string  from  the  point  of  disturbance,  each  part  being  reflected  when 
it  arrives  at  a  fixed  end.  Hence,  (§  53),  the  string  is  thrown  into 


186 


A    MANUAL    OF    PHYSICS. 


segments,  the  wave-length  in  each  case  being  equal  to  twice  the 
length  of  a  loop. 

Thus  the  above  equation  enables  us  to  assert  that  the  funda- 
mental period  of  vibration  of  a  string1  is  directly  jpropor/icmfl/  to 
its  length,  to  the  square  root  of  its  density,  and  to  the  square  root 
of  its  sectional  area,  and  is  inversely  proportional  to  the  square 
root  of  its  tension. 

If  a  string  of  length  I  vibrates  n  times  per  second  (determined  by 
means  of  the  syren),  the  speed  of  the  wave  is  lln. 

The  higher  vibrations  correspond  to  tones  which  are  respectively 
one,  one-and-a-half,  two,  two-and-a-quarter,  etc.,  octaves  above  the 
fundamental  tone<  It  is  interesting  to  compare  this  result  with  the 
corresponding  result  in  the  case  of  a  vibrating  rod  which  is  fixed  at 
both  ejids.  In  the  latter  case,  the  transversely  propagating  force  is 
the  flexural  stress  ;  in  the  present  case,  it  is  the  tension  of  the  cord, 
in  which  flexural  stress  has  no  existence. 

172.  Vibration  of  Air -Columns. — When  a  condensation  reaches 
the  closed  end  of  a  pipe,  or  tube,  containing  air  through  which 
sound  is  travelling,  the  increase  of  pressure  which  takes  place  can 
only  be  relieved  by  expansion  of  the  air  backwards  along  the  pipe. 
Similarly,  when  a  rarefaction  reaches  the  closed  end,  the  consequent 
diminution  of  pressure  can  only  be  changed  by  a  flow  of  the  air 
contained  in  the  pipe  towards  that  end.  In  other  words,  the  wave 
is  simply,  reflected  at  the  closed  end,  the  result  being  that  two 
sound-waves  are  travelling  simultaneously  along  the  pipe,  in  opposite 
directions,  with  equal  speed.  And  the  state  of  pressure  due  to  each 
is  always  in  the  same  phase  at  the  closed  end  of  the  pipe.  But,  at 
a  certain  distance  from  the  closed  end,  an  outgoing  condensation 
meets  an  incoming  rarefaction.  Hence,  on  the  assumption  that  no 
energy  has  been  lost  in  the  process  of  reflection,  the  pressure  has 
here  its  normal  value.  And  this  normal  condition  is  maintained 
constantly  so  long  as  the  wave-length  of  the  disturbance  is  un- 
altered, for  the  two  oppositely  -  travelling  waves  are  always  in 
opposite  phases  as  regards  pressure  at  this  place,  which  is  evidently 
distant  by  one-quarter  of  a  wave-length  from  the  closed  end  of  the 
pipe. 

The  conditions  of  motion  of  the  particles  of  air  are  simply 
reversed  when  the  wave  is  reflected,  and,  consequently,  the  incident 
and  the  reflected  disturbances  are  in  opposite  phases  as  regards 
vibration  when  they  are  in  the  same  phase  as  regards  pressure  or 
density.  Hence  the  places  of  maximum  variation  of  pressure  are 
places  of  no  vibration,  and  the  places  of  uniformly  normal  pressure 
are  those  of  maximum  vibration.  Thus  a  node  occurs  at  the  closed 


SOUND.  187 

end  of  the  pipe,  and  other  nodes  appear  at  equal  intervals  of  half  a 
wave-length. 

It  is  obvious  that  a  loop,  or  place  of  greatest  motion,  must  occur 
at  the  open  end  of  a  pipe.  For,  when  a  condensation  reaches  it, 
the  air  is  free  to  expand  outwards — much  more  free  to  do  so,  in 
fact,  than  to  expand  inwards — and  consequently  the  condensation  is 
succeeded  by  a  rarefaction  which  is  propagated  back  through  the 
pipe.  Similarly,  an  incident  rarefaction  is  filled  up  by  the  influx  of 
the  surrounding  air,  and  so  a  condensation  is  returned.  Thus  we 
see  that  there  is  great  vibration  at  the  open  end  of  the  pipe,  which 
therefore  corresponds  to  a  loop.  [This  might  otherwise  have  been 
determined  by  means  of  the  consideration  that,  in  the  immediate 
neighbourhood  of  the  open  end,  the  air-pressure  is  uniform  and 
equal  to  the  normal  atmospheric  pressure.  Consequently  the  open 
end  is  a  place  of  maximum  vibration.] 

Stopped  and  Open  Pipes. — A  pipe  which  is  closed  at  one  em 
is  called  a  '  stopped '  pipe,  while  one  which  is  open  at  both  ends  is 
called  an  '  open '  pipe. 

In  a  stopped  pipe  the  nodes  evidently  occur  at  the  same  positions 
as  do  the  nodes  of  a  transversely-vibrating  rod  which  is  fixed  at  one 
end.  Thig  is  indicated  in  the  figures  below,  in  which  the  vertical 
distance  between  the  waved  lines  at  any  part  of  the  length  of  the 
pipe  may  be  supposed  to  indicate  the  extent  of  the  vibrations  at  that 
part.  The  first  figure  shows  the  fundamental  mode  of  vibration. 
The  wave-lengths,  and  consequently  the  periods  of  vibration,  in  all 
the  possible  modes  are  evidently  inversely  as  the  odd  numbers  1,  3, 


HC  X  X 

FIG.  82. 

5,  etc. ;  and,  in  the  fundamental  mode  of  vibration,  the  length  of 
the  wave  is  four  times  the  length  of  the  pipe  ;  for  a  condensation 
which  leaves  the  open  end  has  its  phase  reversed  when  it  again 
reaches  that  end  and  is  reflected.  Two  reversals  are  therefore 
necessary  in  order  that  the  original  phase  may  be  attained. 


188  A    MANUAL    OF    PHYSICS.  , 

In  an  open  pipe  the  nodes  occur  at  the  same  positions  as  do  the 
nodes  of  a  transversely-vibrating  rod  which  is  not  fixed  at  either 
end.  The  positions  are  indicated  below.  The  possible  periods  of 
vibration  are  inversely  as  the  natural  numbers  1,  2,  3,  etc.  The 


FIG.  83. 

wave-length  in  the  fundamental  mode  of  vibration  is  twice  the 
length  of  the  pipe ;  for  a  condensation  passing  from  one  end  of  the 
pipe  is  reflected  from  the  other  end  as  a  rarefaction,  and  leaves  the 
original  end  once  more  as  a  condensation.  Hence  an  open  organ- 
pipe  which  has  the  same  fundamental  tone  as  a  given  closed  organ- 
pipe  must  be  twice  as  long  as  the  closed  one  is*. 

Speed  of  Sound. — In  order  to  determine  very  accurately  the  speed 
of  sound  in  any  given  gas  we  merely  require  to  sound  an  organ- 
pipe  of  given  length  which  is  filled  with  that  gas.  By  means  of  the 
syren  the  number  of  vibrations  which  are  made  per  second  is  found. 
And  if  I  be  the  length  of  the  pipe,  while  n  is  the  number  of  vibra- 
tions made  per  second,  the  speed  is  4nl,  or  %nl,  according  as  the  pipe 
is  closed  or  open.  Under  ordinary  atmospheric  conditions  the  speed 
of  sound  in  air  is  about  1,120  feet  per  second. 

173.  Partial  Tones. — Resonance. — From  the  results  of  the  last 
few  sections  it  is  now  evident  that  no  note  given  out  by  a  musical 
instrument  is,  strictly  speaking,  a  pure  tone  ;  in  most  cases  it  very 
obviously  is  not  so.  Thus  the  '  tongue  '  of  a  reed  is  simply  a  rod  or 
strip  of  metal  which  is  fastened  at  one  end,  and  is  caused  to  vibrate 
by  means  of  a  current  of  air  which  is  rendered  intermittent  by  the 
vibrations  which  it  produces  in  the  tongue ;  and  these  air-pulses 
produce  a  note  in  which  occur  tones  corresponding  to  the  various 
forms  of  vibration  which  the  tongue  assumes. 

These  constituent  tones  of  a  note  are  called  the  partial-tones  ; 
and  it  is  usual  to  distinguish  between  the  fundamental  tones  (corre- 
sponding to  the  fundamental  mode  of  vibration  of  the  sounding  body) 
and  the  higher  partial-tones,  which  are  termed  the  overtones. 


SOUND.  189 

Much  (greater  force  is  required  to  set  a  body  in  rapid  vibration 
than  to  set  it  in  slow  vibration,  and  consequently  the  overtones  of  a 
note  are  feebler  than  the  fundamental  tone — each  overtone,  taken 
in  order  of  pitch,  being  weaker  than  the  preceding  one.  In  this 
way  a  note  is  said  to  have  the  same  pitch  as  its  fundamental  tone. 

The  unaided  ear  is  able  to  detect  in  great  measure  the  various 
overtones  which  are  present  in  a  given  note  ;  but,  in  this  analysis, 
it  may  be  greatly  aided  by  means  of  instruments,  the  action  of 
which  depends  upon  the  principle  of  resonance,  according  to  which 
any  sounding  body  can  readily  absorb,  and  give  out  again  of  itself > 
a  sound  wliicli  is  emitted  by  another  sounding  body  which  has  a 
period  of  vibration  identical  ivitli  its  own. 

To  understand  this  principle,  we  need  only  refer  to  a  well-known 
dynamical  analogy :  A  pendulum  of  given  length  has  a  definite 
period  of  vibration,  and  oscillations  of  great  magnitude  may  be 
induced  in  it  by  the  application  to  the  bob  of  the  feeblest  impulses, 
provided  only  that  these  impulses  are  communicated  regularly  at 
instants  the  interval  between  which  is  equal  to  the  natural  period 
of  oscillation  of  the  pendulum.  The  effects  of  all  the  impulses  are 
in  the  same  direction,  and  so  the  total  effect  may  be  very  large 
although  each  single  effect  is  excessively  small.  And,  similarly,  the 
feeble  periodic  impulses  which  are  communicated  through  the 
medium  of  the  air  to  a  body  which  is  capable  of  giving  out  sound 
may,  by  being  properly  timed,  set  that  body  in  such  a  state  of 
vibration  as  to  give  out  the  sound  of  itself  after  the  original  note 
has  ceased. 

The  resonator  of  v.  Helmholtz  consists  of  a  hollow  brass  ball 
with  two  apertures  at  opposite  ends  of  a  diameter.  (This  is  indi- 
cated in  Fig.  84.)  Sound  is  communicated  to  the  air  in  the  ball 
through  the  large  aperture,  and  the  small  aperture  is  applied  to  the 


FIG.  84. 

ear.  The  air  inside  the  ball  has  a  definite  fundamental  period  of 
vibration,  and  therefore  that  tone  (should  such  exist)  which  corre- 
sponds to  this  period  is  very  distinctly  heard — all  others  being 
entirely,  or  almost  entirely,  suppressed. 


190  A   MANUAL   OF   PHYSICS. 

In  many  cases  it  is  desirable  to  intensify  the  fundamental  tone  of 
a  note.  The  principle  of  resonance  shows  that  this  may  be  done 
by  associating  with  the  sounding  body  a  resonating  column  of  air 
which  has  the  same  fundamental  period  of  vibration.  Thus  a  reed 
is  fitted  at  one  end  of  an  open  pipe  of  the  proper  length ;  and  a 
tuning-fork  is  attached  to  a  '  sounding-box,'  which  is  simply  a  closed 
pipe  of  such  a  length  that  the  contained  air-column  has  the  same 
fundamental  period  of  vibration  as  the  tuning-fork  has.  The  box 
itself  is  set  in  vibration  by  the  fork,  and  so  the  motion  is  communi- 
cated to  the  enclosed  air.  The  cavity  of  a  violin  acts  similarly 
as  a  resonator. 

The  sounding-board  of  a  pianoforte  is  thrown  into  vibration  by 
the  wires  which  are  attached  to  it,  and  so  increases  the  mass  of  air 
which  is  made  to  vibrate  when  a  note  is  struck  on  the  instrument. 
Thus,  by  its  resonance,  it  intensifies  the  sound. 

174.  Quality. — There  is  between  two  pure   tones  of  the   same 
pitch  no  difference  of  quality  such  as  that  which  distinguishes  the 
same  note  when  played  on  two  different  instruments.     We  therefore 
conclude  that  difference  of  quality  between  two  notes  is  due  to  the 
existence  of  partial-tones. 

Experiment  shows  that  this  is  actually  the  case,  and  it  indicates 
further  that  the  quality  of  a  note  depends  upon  the  number  of 
partial-tones  which  are  present  in  it ;  that,  the  number  being  the 
same,  it  depends  upon  the  particular  set  of  tones  which  go  to  make 
up  that  number ;  and,  lastly,  that  it  depends  upon  the  relative 
intensities  of  the  partial-tones. 

[It  is  independent  of  the  particular  phase  in  which  any  one  of 
the  constituent  vibrations  may  be  when  the  phases  of  the  others 
are  given.  Now  the  nature  of  the  resultant  vibration  essentially 
depends  upon  this  condition  (§  52,  (1)  ),  and  so  the  above  result 
appears  somewhat  startling.  Its  truth  is  due  simply  to  the  fact 
that  the  human  ear  is  an  instrument  which  resolves  a  compound 
vibration  into  its  separate  constituents,  and  thus  mere  alteration  of 
phase  of  a  constituent  has  no  effect  upon  the  nature  of  the  sound 
which  is  heard.] 

As  a  special  example,  we  may  instance  the  difference  of  quality 
between  two  notes  of  the  same  pitch  sounded  respectively  on  a 
closed  and  an  open  organ-pipe.  In  a  closed  pipe  the  odd  partial- 
tones  alone  occur,  while  in  the  open  pipe  both  the  odd  and  the  even 
tones  are  present. 

175.  Beats. — Consonance  and  Dissonance. — It  has  already  been 
pointed  out  (§  165)  that  the  intensity  of  the  resultant  sound  due 
to  two   component    vibrations   of  slightly   different  period   varies 


SOUN&.  Idi 

periodically  from  a  minimum  to  a  maximum.  These  regularly- 
occurring  maxima  constitute  beats.  One  beat  occurs  in  the  time 
in  which  one  vibration  gains  a  complete  period  upon  the  other ; 
and,  consequently,  the  number  of  beats  which  occur  per  second  is 
equal  to  the  difference  of  the  number  of  vibrations  which  take  place 
per  second  in  the  two  component  tones. 

When  the  beats  succeed  each  other  with  too  great  rapidity,  the 
ear  becomes  unable  to  distinguish  them  from  one  another,  but  it 
still  recognises  a  cKstinct  discontinuity  in  the  sound,  which  produces 
a  harsh  effect  known  as  dissonance. 

The  number  of  beats  which  occur  per  second  when  two  given 
pure  tones  are  sounded  depends,  of  course,  upon  the  absolute,  as 
well  as  upon  the  relative,  pitch  of  these  tones.  When  the  tones  are 
in  the  neighbourhood  of  the  middle  C,  maximum  dissonance  is  pro- 
duced when  they  differ  in  pitch  by  about  half  a  tone ;  and  there  is 
almost  no  dissonance  when  the  interval  is  a  minor  third. 

On  the  other  hand,  beating  may  occur  between  the  partial-tones 
of  two  notes,  and  the  result  is  that  there  may  be  considerable  dis- 
sonance between  the  two,  even  although  their  fundamental  tones 
differ  by  more  than  a  minor  third.  Indeed,  from  this  cause,  none 
of  the  intervals  below  the  octave  form  a  perfect  concord.  Still,  it  is 
only  in  the  cases  of  the  second,  and  the  major  and  minor  sevenths, 
that  the  result  is  actually  classed  as  a  discord.  In  some  cases  the 
greatest  dissonance,  as  above  denned,  does  not  produce  the  most 
disagreeable  effect  upon  the  ear. 

176.  Combination  Tones. — Dissonance  of  Pure  Tones. — When 
the  rapidity  of  the  beats  is  sufficiently  great,  the  effect  upon  the  ear 
is  that  of  a  musical  note  the  pitch  of  which  is  the  same  as  that  of 
a  tone  in  which  the  number  of  vibrations  per  second  is  equal  to  the 
difference  (or  sum)  of  the  number  of  vibrations  per  second  in  the 
two  tones  which  are  producing  the  beats.  Such  tones  are  called 
combination  tones. 

Combination  tones  of  a  higher  order  may  be  produced  between 
the  first  combination  tone  and  either  of  the  primary  tones,  and 
so  on. 

Beating  may  occur  between  a  combination  tone  and  a  primary,  or 
between  two  combination  tones.  The  result  is  that  dissonance  may 
occur  in  the  case  of  two  pure  tones  where  the  interval  is  greater 
than  a  minor  third.  Thus,  when  the  interval  is  a  major  seventh, 
the  higher  tone  will  make  450  vibrations  per  second  if  the  lower 
makes  240  per  second.  The  combination  tone  has  therefore  210 
vibrations  per  second,  which  produces  30  beats  per  second  with  the 
lower  primary. 


CHAPTEE  XV. 

LIGHT  :    INTENSITY,    SPEED,    THEORIES. 

177.  Rectilinear  Propagation.  Intensity.  —  In  a  homogeneous 
medium,  light,  in  general,  moves  in  straight  lines.  The  fact  that 
the  shadow  of  an  object  which  is  cast  on  the  ground  by  sunlight  is 
not  defined  with  mathematical  accuracy,  but  has  a  more  or  less 
blurred  edge,  does  not  disprove  the  statement.  .  The  indistinctness 
of  the  boundaries  of  the  shadow  is  due  to  the  finite  size  of  the  sun's 
disc,  the  light  proceeding  from  each  point  of  which  produces  a 
separate  shadow. 

The  cases  in  which  'the  above  rule  is  departed  from  will  be 
discussed  in  Chap.  XVIII. 

The  total  intensity  of  a  luminous  source  is  measured  by  the 
amount  of  light  which  it  emits  per  unit  time,  and  the  intensity  of 
the  light  at  any  given  point  of  the  medium  is  measured  by  the 
quantity  which  falls  per  unit  time  upon  unit  area  taken  perpen- 
dicular to  the  direction  in  which  the  light  moves  at  that  point. 
And,  since  light  moves  out  uniformly  in  all  directions  from  a  point - 
source  in  a  homogeneous  medium,  it  follows  that  its  intensity  varies 
inversely  as  the  square  of  the  distance  from  the  source ;  for  the 
same  total  quantity  is,  at  different  instants,  spread  over  the  surfaces 
of  different  concentric  spheres,  the  areas  of  which  vary  directly  as 
the  squares  of  their  radii.  (We  assume,  of  course,  that  the  medium 
is  one  which  does  not  absorb  the  light  in  its  passage  through  it. 
If  any  absorption  did  occur,  the  quantities  of  light  passing  through 
the  different  concentric  spheres  could  not  be  equal.) 

Instruments  used  for  the  purpose  of  comparing  the  intensities  of 
different  sources  of  light  are  called  photometers.  The  simplest 
form  of  photometer  consists  of  a  sheet  of  paper  upon  which  there 
is  a  grease  spot.  If  the  paper  be  illuminated  from  behind,  the  spot 
appears  bright ;  if  it  be  illuminated  from  the  front,  the  spot  seems 
dark ;  if  it  be  illuminated  to  an  equal  extent  on  both  sides,  the  spot 
vanishes.  Under  the  latter  condition  the  intensities  of  the  sources 
are  inversely  as  the  squares  of  their  distances  from  the  spot. 


LIGHT  :     INTENSITY,    SPEED,    THEORIES.  198 

In  another  form  of  the  instrument  two  grease  spots  are  employed, 
each  being  illuminated  from  behind  by  one  source  alone.  The 
distances  of  the  sources  are  varied  until  the  two  spots  appear 
equally  bright. 

178.  Speed. — In  last  section  we  have  spoken  .of  the  motion  of 
light.  The  use  of  the  term  is  justified  by  the  fact  that  a  flash  of 
light  is  not  simultaneously  seen  by  two  observers  who  are  situated  at 
different  distances  from  the  source.  When  the  distance  between  the 
two  points  of  observation  is  not  very  large — a  few  miles,  say — the 
interval  of  time  which  is  occupied  by  the  light  in  passing  from  one 
point  to  the  other  is  so  small  that  it  cannot  be  measured  except  by 
very  special  means.  But  there  are  two  astronomical  methods  of 
determining  the  speed  of  light  which  do  not  involve  the  measure- 
ment of  a  small  interval  of  time. 

The  first  of  these  is  due  to  Homer,  who  observed  that  the  eclipses  of 
Jupiter's  satellites  do  not  appear  to  recur  at  equal  intervals  of 
time,  and  pointed  out  that  this  would  be  a  necessary  consequence  of 
the  finite  speed  of  light. 

In  order  to  understand  more  clearly  how  this  may  be,  we  may 
take  an  illustration  from  the  phenomena  of  sound.  If  an  observer 
be  situated  at  a  fixed  distance  from  a  point  at  which  a  gun  is  fired 
off  at  equal  intervals  of  one  minute,  he  will  hear  the  report  at  equal 
intervals  of  one  minute.  But  if,  between  two  successive  discharges 
of  the  gun,  he  move  nearer  to  it,  he  will  hear  the  next  report  at  a 
shorter  interval  of  time  than  one  minute  ;  while,  if  he  move  farther 
from  it,  the  interval  will  necessarily  be  greater  than  one  minute. 
(When  applied  to  light  (§  204),  this  principle  is  usually  called 
Doppler's  principle.)  The  speed  of  sound  may  be  determined  from 
the  results  of  two  such  observations.  Let  r  be  the  time  which 
elapses  between  two  successive  discharges,  and  let  t  be  the  interval 
noted  by  the  observer  between  two  successive  reports  when  he  has 
meanwhile  increased  his  distance  from  the  gun  by  the  amount  d. 
Then,  t'  being  the  (unknown)  time  taken  by  the  sound  to  pass  over 
the  distance  d,  we  get  £  =  r-H'>  and  therefore  the  speed  of  sound  is 
given  by  the  quotient  of  d  by  t  -  r. 

Now  the  eclipses  of  Jupiter's  satellites  occur  at  instants  which 
are  very  accurately  calculable  from  known  astronomical  data.  But 
the  observed  instants  at  which  the  eclipses  apparently  take  place  as 
seen  from  the  earth  do  not  coincide  with  the  calculated  instants ; 
and  the  errors  at  different  times  of  the  year  are  (assuming  the  finite 
speed  of  light)  due  to  the  variation  of  the  distance  between  Jupiter 
and  the  earth.  The  greatest  difference  of  apparent  errors  is  the 
time  which  light  takes  to  pass  over  the  greatest  difference  of  dis- 

13 


194  A  MANUAL   OF   PHYSICS. 

tance.  But  the  greatest  difference  of  distance  between  Jupiter  and 
the  earth  is  the  diameter  of  the  earth's  orbit ;  and  so  we  obtain  the 
time  taken  by  light  to  pass  over  this  known  distance. 

The  speed  of  light  as  deduced  by  this  method  is  about  186,000 
miles  per  second. 

The  other  astronomical  method  is  due  to  Bradley.  He  observed 
that  the  fixed  stars  appear  to  describe  small  ellipses  on  the  surface 
of  the  heavens  in  the  course  of  a  revolution  round  the  sun,  each  star 
being  displaced  from  the  centre  of  its  elliptic  path  in  the  direction 
of  the  earth's  motion  in  its  orbit,  and  each  to  the  same  amount. 
He  concluded  that  this  was  due  to  the  finiteness  of  the  speed  of 
light  as  compared  with  the  speed  of  the  earth  in  its  orbit. 

A  simple  illustration  may  make  this  clear.  On  a  still  day,  rain- 
drops fall  vertically  downwards.  But,  if  one  moves  forward  with 
considerable  speed,  they  do  not  seem  to  fall  vertically ;  they 
apparently  fall  in  a  slanting  line,  which  is  inclined  forwards  from  the 
vertical  in  the  direction  of  the  observer's  motion.  And  it  is  evident 
that  the  apparent  velocity  of  the  drops  is  the  resultant  of  their 
actual  velocity  and  a  velocity  equal  and  opposite  to  that  of  the 
observer. 

The  light  which  comes  from  a  star  appears  to  come  in  a  direction 
which  depends  in  the  same  way  upon  the  velocity  of  light  and  the 
velocity  of  the  earth  in  its  orbit.  This  latter  velocity  and  the 
maximum  angular  displacement  of  a  star  from  its  true  position 
being  known,  we  can  calculate  the  speed  of  light.  The  value  which 
is  obtained  by  this  method  agrees  very  closely  with  that  obtained 
by  Eomer's  method. 

Fizeau  was  the  first  (1849)  to  determine  the  speed  of  light 
by  direct  experiment.  He  caused  a  beam  of  light  to  pass  out 
through  the  gap  between  two  of  the  teeth  of  a  toothed  wheel,  the 
teeth  and  gaps  of  which  were  all  of  one  size.  This  beam  was 
reflected  from  a  mirror,  placed  at  a  distance  of  a  few  miles  from  the 
wheel,  in  such  a  way  that  it  passed  back  again  through  the  same 
gap  between  the  teeth  of  the  wheel.  The  wheel  was  then  caused  to 
rotate,  and,  at  a  certain  rate  of  rotation,  it  was  found  the  light 
ceased  to  pass  back  between  the  teeth ;  the  reason  being  that  an 
adjacent  tooth  had  moved  into  the  place  of  the  gap  in  the  time  that 
the  light  took  to  travel  twice  over  the  distance  between  the  wheel 
and  the  mirror.  The  rate  of  rotation  of  the  wheel,  and  the  number 
of  teeth  which  it  contained,  being  known,  the  time  which  was  taken 
by  the  light  to  pass  over  the  given  distance  can  be  readily  found. 
If  N  is  the  number  of  revolutions  which  it  made  per  unit  of  time, 
while  n  is  the  total  number  of  gaps  and  teeth  in  its  circumference, 


LIGHT  :     INTENSITY,    SPEED,    THEORIES.  195 

the  speed  of  light  is  2dNn,  d  being  the  distance  between  the  wheel 
and  the  mirror. 

Fizeau's  experiments  were  repeated  some  time  afterwards  by 
Cormi  with  improved  apparatus ;  and,  still  more  recently,  a  further 
improvement  of  the  same  method  has  been  effected  by  Professor 
G.  Forbes  and  Dr.  J.  Young. 

In  Foucault's  method  (recently  improved  by  Michelson),  a  beam 
of  light,  after  passing  through  a  slit,  falls  on  a  mirror  which  can  be 
made  to  rotate  about  an  axis  parallel  to  the  slit.  After  reflection 
from  the  mirror,  the  light  passes  through  a  lens,  which  brings  it 
to  a  focus  on  a  fixed  mirror.  This  mirror  being  so  placed  as  to 
exactly  reverse  the  course  of  the  beam,  the  light  once  more  falls 
upon  the  first  mirror,  and  is  reflected  from  it.  If  the  latter  is 
rotating,  and  has  turned  through  a  sensible  angle  in  the  time 
taken  by  the  light  to  pass  twice  over  the  distance  between  it  and 
the  other,  the  beam  will  not  pass  back  through  the  slit,  but  will  be 
deflected  from  it  through  a  measurable  distance.  The  speed  of 
light  may  be  found  in  terms  of  the  two  distances  just  mentioned, 
together  with  the  rate  of  rotation  of  the  revolving  mirror  and  the 
distance  between  it  and  the  slit. 

These  two  experimental  methods  give  values  of  the  speed  of  light 
which  agree  very  closely  with  the  values  obtained  by  the  two 
astronomical  methods. 

Foucault's  method  is  so  sensitive  that  it  may  be  used  successfully 
when  the  distance  between  the  mirrors  is  only  a  few  feet,  and  it 
lends  itself  readily  to  the  determination  of  the  speed  of  light  in 
different  media,  such  as  glass,  water,  etc.  The  speed  is  found  to  be 
less  in  dense,  than  in  rare,  media. 

179.  Theories. — The  transference  of  light  involves  motion  of 
matter ;  for  when  light  is  absorbed  by  any  body,  increased  motion  J* 
of  the  particles  of  that  body  is  generally  produced.  Indeed,  in  the 
radiometer  (§  153),  visible  motion  of  a  considerable  mass  of  matter 
may  follow  the  absorption  of  light.  Another  marked  example  will 
be  found  in  §  376. 

Hence  we  see  that  transference  of  light  implies  transference  of 
energy ;  and  it  is  in  this  sense  that  we  speak  of  light  as  a  form  of 
energy. 

We  are  therefore  limited  to  two  suppositions  regarding  the  physical 
nature  of  light.  It  may  consist  in  the  actual  propagation  of 
material  particles,  or  corpuscles,  from  the  luminous  object ;  or,  it 
may  consist  in  the  propagation  of  wave-motion  through  a  material 
medium  which  fills  space, 

13—2 


196  A   MANUAL   OF   PHYSICS. 

The  former  theory  is  known  as  the  Corpuscular  Theory,  the 
latter  as  the  Wave  Theory,  or  Undulatory  Theory,  of  light. 

If  the  corpuscular  theory  were  true,  the  mass  of  a  corpuscle  must 
be  excessively  small.  For  vision,  according  to  this  theory,  is  due  to 
the  impact  of  the  corpuscles  upon  the  retina;  and  the  speed  of 
these  corpuscles  is  so  great  that,  unless  their  individual  mass  were 
almost  vanishingly  small,  the  structure  of  the  eye  would  be 
completely  destroyed  by  the  impact.  The  theory  is  met  by  a 
number  of  difficulties  at  the  very  outset.  Thus  it  is  somewhat 
difficult  to  account  for  the  fact  that  the  corpuscles  have  the  same 
speed  whatever  be  the  temperature  of  the  object  from  which  they  are 
projected.  Again,  the  mass  of  a  luminous  body  must  be  appreciably 
affected  by  the  emission  of  particles ;  but  there  is  no  evidence  of 
any  such  effect.  Still,  if  we  boldly  overlook  any  such  preliminary 
difficulties,  we  shall  find  that  the  theory  enables  us  to  account 
readily  for  many  of  the  phenomena  of  light,  although  ultimately  it 
fails  us  altogether. 

On  the  wave-theory,  vision  is  due  to  the  communication  of  the 
vibrations  of  the  assumed  luminiferous  medium  (called  the  ether)  to 
the  nerve-ends  of  the  retina.  The  molecules  of  a  luminous  body  are 
(§  202)  in  rapid  vibratory  motion,  and  this  motion  is  communicated 
to  the  particles  of  the  ether,  and  is  propagated  through  it  from  par- 
ticle to  particle  giving  rise  to  a  series  of  waves  which  travel  with  the 
speed  of  light.  The  investigation  of  §  161  has  a  direct  applica- 
tion to  the  present  case,  and  shows  that  the  intensity  of  light  is  pro- 
portional to  the  square  of  the  amplitude  of  vibration  of  the  particles 
of  the  medium,  and  that  the  energy  of  the  medium,  when  light 
passes  through  it,  is  one-half  kinetic,  one-half  potential.  (On  the 
corpuscular  theory  the  intensity  must  be  proportional  to  the  space - 
density  of  the  corpuscles.)  We  shall  find  subsequently  (Chap.  XIX.) 
that  the  direction  of  vibration  in  the  medium  must  be  perpendicular 
to  the  direction  of  propagation  of  the  waves. 

The  wave-length  is  the  distance,  measured  in  the  direction  of 
propagation,  from  any  point  to  the  next  point  where  the  motion  is 
similar.  (Compare  §  157.) 

The  wave-theory  is  not  without  its  difficulties — many  of  them, 
indeed,  are  of  a  most  formidable  nature.  But,  as  will  appear,  the 
evidence  in  favour  of  it  is  of  such  an  overwhelming  nature  that  we 
now  practically  regard  its  truth  as  definitely  established.  Newton 
rejected  it  because  he  was  unable  to  explain  by  it  the  rectilinear 
propagation  of  light.  We  now  know  that  the  existence  of  rays 
is  a  necessary  consequence  of  the  fundamental  principles  of  the 
theory. 


LIGHT  :   INTENSITY,  SPEED,  THEORIES.  197 

180.  Colour. — Many  different  kinds  of  light  are  recognised.  We 
speak  of  red  light,  blue  light,  etc.  On  the  corpuscular  theory  the 
difference  must  be  inherent  in  the  corpuscles.  On  the  wave-theory 
the  difference  is  a  mere  difference  of  wave-length,  or  of  vibrational 
period — which  is  only  another  way  of  stating  the  same  thing,  since 
the  speed  of  propagation  of  light  of  all  colours  has,  in  free  space, 
one  definite  value  only. 


CHAPTER  XVI. 

LIGHT  :    REFLECTION,    REFRACTION,    DISPERSION. 

181.  Laws  of  Reflection. — When  a  ray  of  light  reaches  the  bound- 
ing surface  of  a  homogeneous  medium  through  which  it  is  passing, 
it  is,  in  part  at  least,  bent  back  or  reflected,  and  pursues  a  different, 
though  still  rectilinear,  path. 

The  reflected  and  incident  rays  lie  in  one  plane  with,  and  make 
equal  angles  with,  the  normal  to  the  surface. 


FIG.  85. 

Let  EBF  (Fig.  85)  represent  a  section  of  the  bounding  surface  by 
the  plane  of  the  paper,  and  let  BD  be  the  normal  to  the  surface  at 
the  point  B  whereon  the  incident  ray  AB  falls.  Then,  BC  being 
the  reflected  ray,  the  angles  i  and  r,  which  AB  and  BC  make  with 
BD,  are  equal,  and  AB,  BC,  and  BD  all  lie  in  one  plane,  which  is 
normal  to  the  reflecting  surface.  The  angles  i  and  r  are  called, 
respectively,  the  angle  of  incidence  and  the  angle  of  refraction. 

Many  surfaces,  such  as  those  of  chalk  or  of  rough  white  paper, 
scatter  the  incident  light  in  all  directions.  But  this  is  merely  a 
special  case  of  reflection.  At  every  point  of  such  a  surface  rays  are 
reflected  in  accordance  with  the  above  law ;  but  the  whole  surface 
is  practically  made  up  of  an  excessively  great  number  of  very  small 
planes,  which  are  indiscriminately  inclined  in  all  possible  ways. 

The  intensity  of  the  reflected  ray  depends  upon  the  angle  of  inci- 
dence, being  greatest  when  the  angle  is  large,  and  having  its  least 


LIGHT  :     REFLECTION,    REFRACTION,    DISPERSION. 


199 


value  when  the  incidence  is  perpendicular.  It  varies  (the  intensity 
of  the  incident  ray  being  fixed)  with  the  nature  and  the  state  of 
surface-polish  of  the  reflecting  substance,  and  it  depends  also  upon 
the  nature  of  the  medium  through  which  the  light  is  travelling. 

182.  Reflection  from  Plane   Surfaces. — If  a  ray   of    light  be 
emitted  from  a  point  B  (Fig.  86)  and  reach  a  point  A,  after  reflection 


D 


FIG.  86. 

from  a  plane  surface,  CD,  the  actual  length  of  the  path  APB  is  the 
shortest  possible  consistent  with  the  condition  of  reflection  at  the 
given  surface. 

For  an  eye  placed  at  the  point  A  will  see  the  light  in  the  direction 
AP  as  if  it  came  from  a  point  B',  which  is  situated  on  the  normal 
drawn  from  B  to  the  surface.  [This  is  so  since  the  eye  sees  an 
object  by  means  of  a  cone  of  rays  :  and  the  angle  of  the  cone  is  un- 
altered by  reflection  since  (Fig.  87)  the  angles  which  a^pi  and  a.2p.2 


FIG.  87. 

make  with  the  reflecting  plane  are  respectively  equal  to  the  angles 
which  Pid'i  and^2a'2,  the  continuations  of  the  lines  bpi  and  bp.2,  make 
with  that  plane.  And  so  the  continuations  of  a^  and  a.2p2  meet  at 
a  point  &',  which  is  such  that  bpib'  and  bp.2b'  are  both  bisected  by  the 
surface  ;  and  therefore  b'  and  6  lie  on  the  same  normal  to  the  surface, 
and  are  equally  distant  from  it.]  And  any  other  path,  AP'B,  being 
equal  to  AP'B',  is  greater  than  APB,  which  is  equal  to  APB'. 

The  point  B'  is  called  the  image  of  the  point  B.  If  B  were  a  body 
of  finite  size,  each  point  of  it  would  give  rise  to  an  image  ;  and  the 
whole  congeries  of  these  point-images  constitutes  the  image  of  the 
body  B. 


200 


A   MANUAL    OF   PHYSICS. 


183.  Reflection  from  Curved  Surfaces. — Let  Q  (Fig.  88)  be  the 
section  of  a  spherical  mirror  by  the  plane  of  the  paper.  Let  0  be 
the  centre  of  the  sphere,  and  let  a  ray,  UP,  emitted  by  a  luminous 


U 


FIG.  88. 

object  at'the  point  U,  be  reflected  to  the  point  V.  We  have  PVQ  = 
QPV+  PUQ  =  2UPO  +  PUQ.  Therefore  PVQ  +  PUQ  =  2(UPO  + 
PUQ)  =  2POQ.  When  P  and  Q  are  nearly  coincident  this  becomes 
approximately  PQ/PV+PQ/PU  =  2PQ/PO  or  1/PV+1/PU  =  2/PO. 
If  we  denote  by  u,  v,  and  r,  the  lengths  of  the  lines  PU,  PV,  and 
PO  respectively,  this  gives 


This  equation  enables  us  to  calculate  the  position  of  the  image  V 
when  the  position  of  the  object  U  is  given.  If  U  be  situated  at 
infinity  towards  the  left-hand  side  of  the  diagram,  V  is  half-way 
between  0  and  Q.  This  point  is  called  the  principal  focus  of  the 


FIG.  89. 

mirror.  As  U  moves  in  from  infinity  V  moves  out  to  meet  it,  and 
the  two  points  coincide  at  O,  the  centre  of  the  sphere.  The  positions 
of  U  and  V  are  now  interchanged,  and  finally,  when  U  is  at  the 
principal  focus,  V  is  at  infinity  towards  the  left.  Whenever  U 
comes  nearer  Q  than  the  distance  of  a  semi-radius,  the  quantity  v 
becomes  negative,  that  is,  the  image  passes  away  (Fig.  89)  behind 
the  mirror,  and  gradually  approaches  it  from  infinity  in  this  direction 
until  both  object  and  image  coincide  at  Q. 


LIGHT  :     REFLECTION,    REFRACTION,    DISPERSION. 


201 


In  all  cases  V  and  U  are  interchangeable,  that  is  to  say,  V  may  be 
the  object  and  U  will  then  be  the  image.  A  slight  inspection  of  the 
two  diagrams  will  make  this  clear. 

The  image,  when  on  the  same  side  of  the  mirror  as  the  object  is 
on,  is  called  a  real  image  ;  when  on  the  opposite  side,  it  is  called  a 
virtual  image. 

It  is  evident  that  an  object  on  the  convex  side  of  the  sphere  can 
have  a  virtual  image  only.  In  this  case  the  above  formula  becomes 

l_l=2f 
v     u     r 

which  is  the  modification  of  the  formula  necessary  to  make  it  apply 
to  the  case  of  reflection  from  a  convex  spherical  mirror. 


FIG.  90. 

The  law — stated  in  last  section  with  reference'to  a  plane  mirror — 
that  light  takes  the  shortest  possible  path  between  two  points  con- 
sistent with  the  condition  of  reflection  at  the  given  surface,  still  holds 
in  the  case  of  any  surface  provided  that  we  limit  the  statement  to 
other  paths  which  do  not  finitely  differ  from  the  actual  path  of  the 
light.  The  necessity  for  this  limitation  will  be  evident  if  we  consider 


FIG.  91. 

that  light  diverging  from  a  focus  of  a  reflecting  ellipsoid  may  take 
the  longest  possible,  as  well  as  the  shortest  possible,  path. 

Figs.  90  and  91  show  positions  of  real  and  virtual  images  :  a  and 
a'  are  mutually  real  images ;  b  and  bf  are  mutually  virtual  images ; 
b  being  the  image  of  b'  in  a  convex  mirror,  while  a,  a',  and  b'  are 


202 


A   MANUAL    OF    PHYSICS. 


the  images  of  a',  a,  and  6  in  a  concave  one.  When  formed  by  one 
reflection,  or  by  an  odd  number  of  reflections,  a  real  image  is  inverted, 
but  a  virtual  image  is  not  inverted.  The  positions  of  the  various 
points  of  the  image  corresponding  to  given  points  of  the  object  are 
found  by  means  of  the  above  formulae. 

184.  Caustics  :  Focal  Lines. — Let  CBQ  (Fig.  92)  represent  the 
section  of  a  spherical  mirror  by  the  plane  of  the  paper,  and  let  PC, 


FIG.  92. 

PB  represent  two  rays  which,  diverging  from  the  point  P,  fall  upon 
the  mirror  :  let  also  the  reflected  rays,  Cpf  and  Bp/,  intersect  in  the 
point  p. 

Since  the  vertical  angles  at  the  intersection  of  Cp  and  OB  (a 
radius)  are  equal,  we  have 

OCp+COB  =  OBp+CpB,  or  OCP+COB 
This  gives       COQ  -  CPO+COB=BOQ  -BPO+CpB, 


whence  CPB+C#B 

[The  results  of  last  section  follow  as  a  particular  case  of  this.] 
Let  CB  be  an  infinitesimally  small  arc  of  constant  length,  and  let 
T  be  the  point  at  which  the  tangent  from  P  meets  the  circle  CBQ. 
CPB  always  diminishes,  and  therefore  (by  the  above  equation)  CpB 
constantly  increases  as  C  moves  from  Q  towards  T.  Hence  the 
length  of  Cp  always  diminishes  as  its  inclination  to  CO  increases, 
until  finally  p  coincides  with  T. 

The  locus  of  p  is  called  the  caustic  curve,  and  is  indicated  by  the 
dotted  curve  in  the  figure.  It  touches  the  circle  at  the  point  T  and 
the  line  PQ  at  a  point  m,  the  position  of  which  may  be  found  by  the 
formula  of  last  section. 


LIGHT  :     REFLECTION,    REFRACTION,   DISPERSION.  203 

If  the  whole  figure  be  rotated  about  the  line  PQ,  the  circle  CBQ 
traces  out  the  spherical  reflecting  surface,  and  the  caustic  curve 
traces  a  continuous  surface  which  is  called  the  caustic  surface.  All 
points  on  this  surface  are  more  intensely  illuminated  by  the  reflected 
light  than  any  point  which  does  not  lie  upon  it,  and  the  cusp  (at  ra) 
is  the  place  of  most  intense  illumination.  All  rays  which  are 
reflected  from  the  surface  pass  through  the  line  PQ. 

Let  us  suppose  that  a  small,  but  finite,  circular  cone  of  rays  falls 
upon  the  reflecting  surface  in  the  neighbourhood  of  the  points  B,  C. 
All  rays  from  points  on  the  small  circle,  the  pole  of  which  is  Q,  and 
which  passes  through  B,  intersect  PQ  in  the  point  /;  and  all  rays 
from  points  on  the  similar  circle  through  C  pass  through/';  and  so 
on.  It  is  evident,  therefore,  that  a  plane  which  is  perpendicular  to 
the  axial  line  of  the  reflected  cone,  and  which  passes  through  the 
point  in  which  the  axial  line  intersects  the  line  PQ,  will  cut  the  cone 
in  an  elongated  figure-of-eight-shaped  area,  which  may  be  regarded 
as  a  straight  line,  and  is  called  the  secondary  focal  line. 

Again,  a  plane  drawn  perpendicular  to  the  axial  line  through 
the  point  in  which  that  line  touches  the  caustic  surface,  cuts  this 
surface  in  a  circle,  and  all  the  reflected  rays  will  pass  through  the 
plane  in  the  immediate  neighbourhood  of  a  small,  practically 
straight,  portion  of  the  circle  ;  so  that  there  is  another,  nearly 
linear,  normal  section  of  the  reflected  cone.  This  is  called  the 
primary  focal  line. 

The  two  focal  lines  are  mutually  perpendicular. 

An  approximately  circular  section  exists  between  the  two  linear 
sections.  This  is  called  the  circle  of  least  confusion,  and  is  the 
place  where  the  reflected  light  most  nearly  converges  to  a  point. 

185.  The  law  of  reflection  follows  readily  from  the  principles  of 
the  corpuscular  theory.  Let  pq  (Fig.  93)  represent  the  path  of  a 


FIG.  93. 

corpuscle,  and  let  AB  be  the  surface  from  which  the  corpuscle  is 
reflected.  When  the  corpuscle  comes  within  a  certain  small  distance 
from  the  surface,  indicated  by  the  line  ab,  it  experiences  the 
attraction  of  the  medium  which  is  bounded  by  the  surface  AB,  and 
so  is  bent  from  its  rectilinear  path.  The  mutual  action  of  the  par- 


204  A   MANUAL   OF   PHYSICS. 

ticle  and  the  medium  may  alternate  from  attraction  to  repulsion 
many  times  according  to  an  unknown  law,  but  it  must  ultimately 
be  a  repulsion  which  (at  r)  stops  the  motion  of  the  corpuscle  towards 
the  surface.  The  mutual  forces  still  acting  as  before,  the  particle 
must  now  describe  a  path  rq'  precisely  similar  to  rq,  until,  at  qr, 
being  freed  from  the  action  of  the  reflecting  medium,  it  describes  a 
rectilinear  path,  q'p',  which  is  inclined  to  AB  at  the  same  angle  as 
pq  is.  And,  since  the  action  of  the  medium  is  everywhere  in  lines 
perpendicular  to  the  surface  AB,  the  particle  retains  its  velocity 
parallel  to  AB  unaltered  during  the  process  of  reflection,  and 
the  lines  pq  and  p'q'  are  in  one  plane  with  the  normal  to  the 
surface. 

186.  The  wave-theory  also  affords  a  ready  explanation  of  the 
phenomena  of  reflection. 

But,  before  dealing  with  this  point,  it  is  necessary  to  consider  the 
explanation,  on  this  theory,  of  the  rectilinear  propagation  of  light. 
Newton  did  not  see  how  to  account  for  it,  and  so  supported  the  cor- 
puscular theory.  Huyghens  was  the  first  to  show  that  the  wave- 
theory  furnishes  a  ready  explanation  of  the  phenomenon. 


FIG.  94. 

Let  AB  (Fig.  94)  represent  a  portion  of  a  spherical  wave-front 
diverging  from  the  point  0.  All  points,  such  as  a,  6,  c,  on  this  sur- 
face become  centres  of  disturbance  from  which  secondary  spherical 
waves  diverge.  With  radius  AA',  or  BB',  equal  to  the  distance 
which  light  will  travel  in  a  certain  time,  t,  describe  circles  from 
a,  b,  c  as  centres.  These  circles  will  all  touch  another  spherical 
surface,  A'B',  concentric  with  AB.  This  constitutes  the  new  wave- 
front.  At  all  points  of  this  surface  secondary  wavelets  are  super- 
posed, and  so  a  strong  resultant  effect  may  be  produced.  At  no 
other  points,  besides  those  on  A'B',  are  the  effects  of  the  separate 
wavelets  superposed,  and  the  isolated  wavelets  are  too  feeble  to  pro- 
duce light.  Hence  the  rays  included  in  the  region  AOB  diverge 
outwards  in  straight  lines. 

The  above  explanation  is  due  to  Huyghens,  and  will  suffice  for 


LIGHT  :     REFLECTION,    REFRACTION,    DISPERSION.  205 

our  present  purpose.  But,  as  Fresnel  pointed  out,  Young's  principle 
of  interference  is  essential,  in  addition  to  the  above,  in  order  to 
make  the  demonstration  rigorous.  See  §  226. 

We  shall  assume  for  the  sake  of  simplicity  that  we  are  dealing 
with  a  plane  wave  propagated  through  the  luminiferous  medium. 
Let  ADF  (Fig.  95)  represent  the  reflecting  surface,  and  let  the  given 
disturbance  have  reached  the  position  ABC  ;  so  that  ABC  represents 
a  portion  of  the  plane  wave-front,  to  which  the  rays  (of  which  three 
are  indicated  in  the  figure)  are  everywhere  perpendicular,  the 
medium  being  assumed  to  be  homogeneous  and  isotropic. 


When  the  wave  reaches  the  point  A,  the  particles  of  the  ether  at 
that  point  are  set  in  vibration  and  give  rise  to  a  spherical  wave 
which  spreads  out  from  A  as  centre.  If,  from  A,  we  draw  a  sphere 
with  radius  AP  =  CF,  we  get  the  position  of  this  spherical  wave 
when  the  disturbance  originally  at  C  has  reached  the  reflecting  sur- 
face. Similarly,  DE  being  parallel  to  ABC,  if  we  draw  from  D  as 
centre  a  sphere  with  radius  DQ  =  EF  we  get  the  corresponding  posi- 
tion of  the  spherical  wave  which  is  originated  at  the  point  D  when 
the  wave-front  reaches  it.  All  such  spheres  touch  a  plane  surface, 
PQF,  which  is,  therefore,  the  wave-front  after  reflection.  Thus  we 
see  that  a  plane  wave  remains  a  plane  wave  after  reflection. 

But,  further,  AP  =  CF,  and  AF  is  common  to  the  two  right-angled 
triangles  ACF  and  APF.  Therefore  the  angles  CAF  and  PFA  are 
equal,  that  is,  the  reflected  wave-front  has  the  same  angle  of  inclina- 
tion to  the  reflecting  surface  as  the  incident  wave-front  has.  And 
this — since  the  rays  are  perpendicular  to  the  wave-fronts — gives  the 
known  law  of  reflection. 

187.  Laws  of  Refraction. — A  ray  of  light,  on  passing  from  one 
medium  into  another  of  different  density,  is  in  general  bent  from  its 
original  direction,  and  is  said  to  be  refracted.  The  angle  which 
the  refracted  ray  makes  with  the  normal  is  called  the  angle  of 
refraction. 

The  refracted  and  incident  rays   lie  in  one  plane  with   the 


206 


A   MANUAL    OF    PHYSICS. 


normal   to  the  surface  and  the  sine   of  the   angle   of  incidence 
bears  a  constant  ratio  to  the  sine  of  the  angle  of  refraction, 

The  angles  of  incidence  and  refraction  (Fig.  96)  being  denoted 
by  the  letters  i  and  r  respectively,  the  above  law  may  be  written 


The  constant,  /i,  is  called  the  index  of  refraction.  When  the 
refraction  takes  place  from  a  less  dense  into  a  more  dense  medium, 
H  is  generally  greater  than  unity  ;  and,  conversely,  /*  is  usually  less 
than  unity  when  the  light  passes  from  a  more  dense  into  a  less 
dense  medium. 


FIG.  96. 


FIG.  97. 


The  intensity  of  the  refracted  ray  depends  upon  the  angle  of  inci- 
dence. It  is  greatest  when  the  incident  ray  is  perpendicular  to  the 
refracting  surface,  and  diminishes  as  the  angle  increases.  In  the 
case  of  refraction  into  a  denser  medium,  minimum  intensity  is 
attained  at  grazing  incidence  :  in  the  case  of  refraction  into  a  rarer 
medium,  the  minimum  is  reached  when  the  angle  of  incidence  is 
less  than  90°  ;  and  at  higher  angles  of  incidence  no  refraction  occurs. 

188.  Refraction  through  a  Plane  Surface.  —  Let  a  ray,  aO  (Fig. 
97)  fall  upon  the  plane  surface,  AB,  of  a  medium,  the  refractive 
index  of  which  is  ju  —  that  of  the  first  medium  being  taken  as  unity. 
If  we  assume  p  to  be  greater  than  unity,  the  path  of  the  ray  in  the 
second  medium  will  be  a  line  06,  such  that  sin  aOc  =  p  sin  bQd,  CD 
being  perpendicular  to  AB  at  the  point  of  incidence  0.  Also  light 
travelling  in  the  direction  60  will  emerge  into  the  first  medium  in 
the  direction  Oa,  which  is  such  that 


sin  60d  =     sin  aOc. 

/' 
Now  suppose  ^00  =  90°.     The  light  will  enter  the  second  medium 


LIGHT:   REFLECTION,  REFRACTION,  DISPERSION. 


207 


in  a  direction  Oc,  such  that  sin  cOD=//;  and,  conversely,  light 
travelling  in  the  direction  cO  will  emerge  into  the  first  medium 
so  as  just  to  graze  along  the  surface.  Light  passing  in  the  direction 
eO,  where  eOD  >  cOD,  cannot  enter  the  first  medium  at  all,  but  will 
suffer  total  reflection  in  the  direction  oe',  in  accordance  with  the 
ordinary  law.  (This  proves  the  concluding  remark  of  last  section.) 
The  ray  eo  is  said  to  suffer  Total  Reflection,  and  the  limiting  angle 
cOT>  is  called  the  Critical  Angle. 

The  refractive  index  of  water,  relatively  to  that  of  air,  is  about 
4/3  for  ordinary  yellow  light.  Hence  an  eye  placed  underneath  the 
surface  of  water  will  see  objects  above  the  surface  by  means  of  rays 
which  are  crowded  into  a  cone,  the  sine  of  the  semi-vertical  angle  of 
which  is  3/4. 

The  angle  of  deviation  of  a, ray  from  its  original  direction  by  a 
single  refraction  is  i-r,i  and  r  being  respectively  the  angles  of 
incidence  and  refraction. 


FIG.  98. 

From  0  (Fig.  98)  as  centre  describe  two  circles  APB  and  CQD, 
with  radii  equal  to  unity  and  //  respectively.  Let  COQ  =  r,  and  draw 
QPN  parallel  to  CO.  Then  ON  =  sin  OPN =//  sin  OQN  =/*  sin  r,  and 
so  OPN  =  i. 

Now  OP  and  OQ  are  fixed  in  length,  and  PQ  is  always  parallel 
to  OC  ;  so  that  OPQ  and  OQP  become  more  nearly  right  angles, 
and  PQ  becomes  greater  at  an  increasing  rate  as  i  increases  up  to 
90°.  Hence  i  -  r  increases  faster  and  faster  as  either  i  or  r  increase 
uniformly. 

Next  suppose  that  PQ  moves  out  from  OC  through  an  infinitesimal 
distance  into  a  position  P'Q',  and  let  OQ'  be  then  turned  back  to 
coincide  with  OQ.  P'  will  then  take  a  position  p  such  that  AJO  is 


208 


A  MANUAL   OF   PHYSICS. 


greater  than  AP.  By  this  process  i-r  has  increased  by  the 
amount  Pp,  and  PQO  =  r  has  increased  by  the  amount  £>QP  = 
Pp  cos  t/PQ.  Hence  r  —  (i  —  r)  —  2r~i  has  increased  by  the  angle 
Pp  (cos  -i/PQ  —  1).  This  vanishes  when  cos  i  =  PQ.  But  cos  i=NP, 
and  hence  the  condition  is  NP  =  PQ.  This  implies  AC  <  OA,  that  is, 
fi  <  2.  Again,  the  increase  of  2r  —-i  is  positive  or  negative  according 
as  NP  >  or  <  PQ.  Hence  it  is  always  positive  until  the  limiting 
angle  is  reached,  after  which  PQ  still  increases,  while  NP  diminishes, 
so  that  2r-i  diminishes  continuously.  In  other  words  the  con- 
dition NP  =  PO  or  2  cos  i  =  p  cos  r  (since  NQ  =  /*  cos  r)  indicates  a 
maximum  value  of  2r— i. 

This  condition,  combined  with  sin  i=n  sin  r,  gives 

3  sin2  i  =  4  -  ^2. 

Similarly,  the  angle  3r  -  i  increases  under  the  same  conditions 
by  the  amount  Pp  (2  cos  i/PQ  -  1) ;  and  the  condition  for  a  stationary- 
value  is  2NP  =  PQ  or  3  cos  i=(J>  cos  r,  which  implies  /*  <  3.  As  in 
the  former  case  the  condition  indicates  a  maximum.  From  this 
along  with  sin  i  —  ^  sin  r  we  get 

8  sin2  i  —  9  -  /*2. 

These  results  are  of  importance  in  the  discussion  of  the  primary 
and  secondary  rainbows. 


FIG.  99. 

189.  Focal  Lines:  Caustics. — Let  OPo'  (Fig.  99)  represent  an 
axial  section  of  an  infinitesimal  circular  cone  of  rays  diverging 
from  a  point  P  placed  in  a  dense  medium.  The  section,  by  the 
plane  of  the  paper,  of  the  cone  a'pa,  by  which  the  object  P  is 
actually  seen  by  an  eye  situated  in  the  rare  medium,  has  its  vertex 
at  a  different  point,  p,  which  is  nearer  the  surface  than  P  is. 

Let  i't  r',  and  i,  r,  represent  respectively  the  angles  of  incidence 
and  refraction  (regarded  from  the  rarer  medium)  of  the  rays  a'o'P 


LIGHT  :     REFLECTION,    REFRACTION,    DISPERSION.  209 

and  aoP  respectively.     The  angle  of  the  cone,  whose  vertex  is  at  P. 
is  r'  -r,  and  the  angle  of  that  whose  vertex  is  at^>  is  i'—i. 

To  find  the  relation  between  these  angles  we  may  write  the 
equation  sin  i'  =  p,  sin  r'  in  the  form  sin  (i' —  i+i}  =  n  sin  (r'  —  r-\-r). 
This  gives  sin  (i'—i)  cos  i-f-cos  (i'—i)  smi  =  p  [sin  (r'—r)  cos  r + 
cos  (r' — r)  sin  r] ,  and  so,  remembering  that  i' -i  and  r'—r  are 
vanishingly  small,  we  get 

i' —  i  __    cos  'r_tan  i 
r'  —  r        cos  i    tan  r 

But  i' -i=0or  cos  i[0p,  and  r'-r  =  Oo'  cos  r/OP.     Hence 

OP  _  sin  i  .  cos2  r. 
Op  ~~  sin  r    cos2  * 

Now  let  Op  be  prolonged  to  meet  PQ  (the  line  drawn  through  P 
perpendicular  to  the  surface)  mpf,  and  we  get  Op'  sin  z  =  OP  sin  r; 

so  that  Op  =  0p'  — „— * 

cos-r 

This  formula,  which  will  be  of  use  when  we  deal  with  spherica 
lenses  (§  194),  shows  at  once  (cf.  §  184)  that  no  more  than  two 
finitely  distinct  rays  in  any  given  vertical  plane  section  can  inter- 
sect in  one  point,  p ;  but,  when  the  cone  is  very  small,  all  the  rays 
in  a  given  plane  section  pass  approximately  through  one  point. 

We  have  next  to  consider  the  lateral  divergence  of  the  beam  of 
light.  This  is  obviously  unaltered  by  refraction ;  for,  after  refraction, 
a  ray  remains  in  the  same  plane,  passing  through  the  normal,  which 
contained  it  before  refraction.  Hence,  laterally,  the  rays  diverge 
from  points  on  PQ.  And  thus,  since  the  cone  has  some  lateral 
thickness  at  the  pointy,  we  see  that  the  emergent  light  appears  to 
pass,  at  £>,  approximately  through  a  small  line,  the  length  of  which 
is  perpendicular  to  PQ. 

This  is  the  primary  focal  line. 

The  secondary  focal  line  is  the  intersection  of  the  refracted  cone 
with  a  plane  drawn  perpendicular  to  its  axial  line  and  passing 
through  the  point  in  which  the  axial  line  cuts  the  line  PQ. 

At  perpendicular  incidence,  Op  =  Op'  —  OP//z ;  so  that  the  light 
appears  to  diverge  from  a  point  which  is  closer  to  the  surface  than 
the  actual  point  in  the  ratio  of  1  to  fi.  For  example,  the  depth  of 
water  appears  to  be  only  three-fourths  of  its  true  depth. 

The  existence  of  the  primary  focal  line  implies  the  existence  of 
a  caustic  surface.  We  may  map  out  (Fig.  100)  successive  points, 

14 


210 


A    MANUAL    OF    PHYSICS. 


jPi>  A>»  G*CM  corresponding  to  successive  positions  of  the  point  o,  by 
means  of  the  last  equation  above  together  with  the  condition 
sin  i^n  sin  r.  We  thus  obtain  the  caustic  curve  popiP*of,  which 


FIG.  100. 

touches  QP  at  p$  and  Qo'  at  or.  The  line  Po'  indicates  the  limiting 
position  beyond  which  total  reflection  takes  place.  If  the  caustic 
curve  be  rotated  around  the  line  PQ,  the  caustic  surface  will  be 
described. 

190.  Refraction  through  Parallel  Layers. — A  ray  of  light,  which 
is  bent  out  of  its  original  direction  on  entering  a  refracting  layer 
(Fig.  101)  with  parallel  plane  sides,  is,  on  re-entering  the  original 


\ 


FIG.  101. 

medium  on  the  far  side  of  the  layer,  bent  back  again  into  its  first 
direction.  This  follows  at  once  from  the  symmetry  of  the  arrange- 
ment. 

If  an  object  is  looked  at  perpendicularly  through  such  a  layer  of 
thickness,  t,  its  distance  from  the  eye  is  apparently  diminished  by 
the  amount  £(/*— !)//«.  For,  if  d  be  the  distance  of  the  object  from  the 
layer,  an  eye  situated  in  the  layer  would  see  it  as  if  it  were  at  a 
distance  dp  from  the  layer.  Therefore,  when  the  eye  is  just  at  the 
far  side  of  the  layer,  but  still  inside,  the  apparent  distance  of  the 
object  is  t-\-djji.  And,  if  the  eye  be  now  just  outside  the  layer,  this 


LIGHT  :     REFLECTION,    REFRACTION,    DISPERSION. 


211 


distance  is  decreased  in  the  ratio  of  p  to  unity,  which  proves  the 
statement. 

If  a  beam  of  light  passes  through  three  such  layers  the  total 
deviation  of  the  beam  from  its  original  direction  is  the  same  as  if 


FIG.  102. 

the  intermediate  layer  were  absent.  For,  /tx  and  ju2  being  the  refrac- 
tive indices  of  the  first  and  second  media  with  reference  to  the 
second  and  third  respectively  we  have  (see  Fig.  102) 


sm  ?!  _  sm 
sin  ro     sin 


sin  r,     sm 


sin  r      sin  r. 


sm  ^3 
sin  r2 


But,  since  /<.,  is  the  index  of  the  second  medium  expressed  in  terms 
of  that  of  the  third  as  the  unit,  while  jux  is  the  index  of  the  first 
expressed  in  terms  of  that  of  the  second  as  the  unit,  ^3  is  the 
index  of  the  first  medium  expressed  in  terms  of  that  of  the  third. 

It  follows  that  no  effect  is  produced  by  any  number  of  intermediate 
layers.  For  this  reason  it  is  possible  to  calculate  the  refraction  error 
in  the  apparent  altitude  of  stars,  etc. 

191.  Mirage. — When  a  beam  of  light  passes  iion-perpendicularly 
through  a  medium  composed  of  parallel  layers  of  continuously 


B 


FIG.  103. 

varying  density,  the  direction  of  the  beam  constantly  alters.     Let 
us  suppose  that,  over  the  surface  of  the  ground  AB  (Fig.  103)  there 

14—2 


212 


A   MANUAL    OF    PHYSICS. 


exists  a  stratum  of  air  the  density  of  which  continually  increases 
with  its  distance  from  the  surface.  We  may  suppose  also  that, 
above  the  plane  ab,  the  density  remains  uniform.  An  object,  nm, 
will  be  seen  by  an  eye  situated  at  p  by  means  of  rays  which  enter 
the  non-homogeneous  medium  and  are  then  bent  upwards  until,  re- 
entering  the  uniform  medium,  they  pass  straight  to  p.  The  object 
appears  to  be  in  the  direction  of  m'n',  and  is  obviously  inverted 
since  the  rays  from  the  upper  and  lower  extremities  cross  each  other 
before  reaching  the  eye.  But  the  object  may  also  be  seen  directly 
through  the  medium  above  ab,  and  so  the  impression  of  an  object 
with  an  inverted  reflection  is  produced.  This  is  the  case  of  the 
ordinary  mirage  of  the  desert. 

In  Fig.  104,  AB  again  represents  the  surface  of  the  ground,  and 


FIG.  104. 

the  density  of  the  air  is  suppose  to  decrease  from  below  upwards. 
A  direct  mirage  is  thus  produced. 

In  Fig.  105  the  air  is  supposed  to  be  of  uniform  density  between 
AB  and  ab,  but  is  supposed  to  diminish  continuously  in  density 


a 


FIG.  105. 

above  ab.  This  condition  obviously  gives  rise  to  an  inverted 
image. 

192.  Prisms. — When  the  two  plane  bounding  surfaces  of  a 
medium  are  not  parallel  the  emergent  ray  is  no  longer  parallel  to 
the  incident  ray.  Such  an  arrangement  constitutes  a  prism. 

Let  ABC  (Fig.  106)  represent  the  section,  by  the  plane  of  the 
paper,  of  a  prism  of  some  dense  medium,  such  as  glass.  Let  the 
edge,  B,  of  the  prism  be  perpendicular  to  the  plane  of  the  paper,  and 
let  abed  be  a  ray  which  makes  an  angle  i  with  the  (inward)  normal 


LIGHT  :     REFLECTION,    REFRACTION,    DISPERSION. 


213 


to  the  face  AB  and  an  angle  i'  with  the  (outward)  normal  to  the  face 
BC.     Let  r  and  r'  be  the  corresponding  angles  of  refraction. 

The  deviation  of  the  ray  from  its  original  direction  by  the  first 
refraction  is  i— r;    and  at  the  second   surface  this   deviation  is 


FIG.  106. 

farther  increased  by  the  amount  i'  -rr.  But  (§  188)  the  alteration 
of  i-r,  consequent  on  a  given  alteration  of  i,  is  greater  and  greater 
the  larger  i  or  r  is ;  and  any  increase  or  decrease  of  r  involves  an 
equal  decrease  or  increase  of  r'.  Hence,  if  i  be  greater  than  i',  a 
given  diminution  of  i  produces  a  diminution  of  i  —  r,  which  is  greater 
than  the  simultaneous  increase  of  i'  —r'.  Hence  the  ray  has  mini- 
mum deviation  when  i=i',  that  is,  when  the  ray  passes  through 
the  prism  so  as  to  make  B6  =  Be. 

When  the  angle  of  the  prism  is  less  than  sin  -11//*  the  deviation 
i'  -  r'  may  be  in  the  opposite  direction  to  the  deviation  i—  r,  but  it 
is  easy  to  see  that  the  above  result  is  true  in  all  cases. 

The  total  deviation,  d,  is,  in  the  standard  case  above,  i'+i- 
(r'-fr),  which,  in  the  minimum  position,  becomes  2(i—  r)=%i-a, 
«  being  the  angle  of  the  prism.  If  /*  be  the  refractive  index  of  the 
substance  of  which  the  prism  is  composed,  this  gives 

sin 


sin  «/2 

This  affords  a  ready  means  of  determining  the  refractive  index  of 
the  substance  of  the  prism ;  and,  if  the  prism  be  hollow  and  have  its 
sides  made  of  glass  plates  of  uniform  thickness,  the  refractive  index 
of  any  liquid  placed  in  the  hollow  may  be  found. 

193.  Refraction  through  Spherical  Surfaces :  Lenses.  —  Let 
QA.B  (Fig.  107)  represent  a  part  of  a  spherical  surface  the  centre  of 


214 


A   MANUAL   OF    PHYSICS. 


which  is  at  O  ;  and  let  the  refractive  index  of  the  substance  on  the 
convex  side  of  the  surface  be  ju.  Let  rays  PA,  PB,  diverging  from 
P,  meet  the  surface  in  the  points  A  and  B  respectively.  Suppose 


that  Ap,  Bp  are  the  backward  prolongations  of  the  refracted  rays 
— /i  being  assumed  to  be  greater  than  unity.  Join  PO  and  continue 
the  line  to  meet  the  surface  in  Q. 

If  we  denote  the  angles  PAO,  PBO,  pA.0,  pRO,  BPA,  BpA,  BOA, 
by  the  letters  i',  i,  rf,  r,  $,  $,  9,  respectively,  the  diagram  at  once 
gives  i'—i  =  (j>-0,  and  r '  -  r  =  ty  —  B.  Hence,  when  the  angles  are  so 
small  that  we  may  write  sin  i=i,  cos  i  —  1,  etc.,  we  have  (§  188) 
i'  -i=p  (r'  —  r),  and  therefore 

Gu-1)  0=^-0 (1). 

When  A  coincides  with  Q  and  AB  is  very  small  this  becomes 

?-l=  /*   _   1  . 
OQ      jpQ     PQ 

When  /i=-l,  this  gives  the  first  formula  of  §  183. 

Now  let  the  rays  (Fig.  108),  apparently  diverging  from  the  point 
p,  fall  upon  the  second  spherical  surface  of  the  dense  medium  at  the 


O' 


FIG.  108. 


points  A',  B'.     Let  0'  be  the  centre  of  the  new  surface,  and  let  O'p 
meet  it  at  the  point  Q'.     Denote  the  angles  p'B'O',  _pA'O',  p'B'O', 


LIGHT  :    REFLECTION,   REFRACTION,    DISPERSION.  215 

yA'O'  A'/B',  A'O'B',  by  the  letters  if,  i,  r',  r,  ij/',  6',  respectively. 
Since  A'^B'  =  ;//,  the  figure  -gives  i'-i  =  ^  —  9'  and  r1  -r  =  V  -0'. 
Hence,  when  A'B'  is  very  small  (i//  —  0')  =  /*(4/""^')»  or 

Oi-l)0'  =  ,4-*'  ........  (2) 

[Of  course  (2)  might  have  been  deduced  from  (1)  by  the  substitution 
of  1/u  for  ft.] 

From  (1)  and  (2)  we  get 

(/•-I)  (0-00  =  *'-*  ......  (8) 

When  A  (Fig.  107)  coincides  with  Q,  Ar  (Fig.  108)  coincides  with 
Q'  ;  and  so,  when  the  angles  are  sufficiently  small,  we  may  write  (3) 
in  the  form 


r  and  s  being  the  radii  of  the  two  spherical  surfaces,  while  v  and  u 
are  respectively  the  distances  of  p'  from  Q',  and  of  P  from  Q. 

The  quantity  on  the  left-hand  side  of  equation  (4)  is  constant. 
We  may  therefore  write  (4)  in  the  form 

UI-I  ...........  (5) 

/    v     u 

where  /  is  a  constant,  called  the  principal  focal  distance.  When 
u  is  infinite,  that  is,  when  the  incident  rays  are  parallel,  we  get 
f=v;  in  other  words,  /  is  the  distance  from  Q'  of  the  point  from 
which  originally  parallel  rays  seem  to  diverge  after  passing  through 
the  medium.  This  point  is  called  the  principal  focus.  P  and  p'  are 
called  conjugate  foci. 

A  portion  of  a  medium  bounded  by  two  spherical  surfaces  is 
termed  a  lens. 

In  the  case  just  discussed  we  have  taken  s>r,  and  have  assumed 
that  the  concave  sides  of  the  spherical  surfaces  are  directed  towards 
P.  All  lines  drawn  in  the  direction  PQ  are  regarded  as  positive. 
Lines  drawn  in  the  direction  of  QP  are  therefore  negative. 

Equation  (4)  applies  to  all  lenses.  The  quantity  /is  negative  (1), 
when  s  is  positive  but  less  than  r,  (2)  when  r  is  infinite  and  *  is 
positive,  (3)  when  r  is  negative  and  s  is  positive. 

In  such  cases  the  principal  focus  and  the  source  of  the  parallel 
rays  are  on  opposite  sides  of  the  lens,  which  is  necessarily  thickest 
at  the  middle  (Fig.  109). 

But  /  is  positive  (1)  when  r  and  s  are  both  positive  and  s  >  r, 
(2)  when  s  is  infinite  and  r  is  positive,  (3)  when  s  is  negative  and  r 
is  positive. 


216 


A   MANUAL   OF   PHYSICS. 


In  the  three  latter  cases  the  principal  focus  and  the  source  of  the 
parallel  rays  are  on  the  same  side  of  the  lens,  which  is  now  thinnest 
(Fig.  110)  at  the  middle.  Such  a  lens  diminishes  the  convergence 
or  increases  the  divergence  of  the  incident  rays,  while  a  lens  of  the 


FIG.  109. 

previous  type  diminishes  the  divergence  or  increases  the  convergence 
of  the  incident  rays.     Equation  (3)  makes  this  evident  at  once. 

The  formula  (4)  is  true  of  a  given  small  cone  of  rays  which  falls 
perpendicularly  upon  the  lens  at  its  central  part,  and  it  shows  that, 


FIG.  110. 

after  passing  through  the  lens,  all  such  rays  appear  to  diverge  from, 
or  are  converged  to,  another  perfectly  definite  point.  We  shall  see 
in  next  section  that  the  same  result  holds  in  the  case  of  a  small 
obliquely-incident  pencil  of  rays. 

194.  Lenses :  Oblique  Refraction. — Let  DCBA  (Fig.  Ill)  repre- 
sent a  ray,  which,  entering  the  lens  CBQQ'  at  C,  emerges  at  B  and 
cuts  the  line  OQ  in  A.  (The  letters  O,  0',  Q,  Q'  have  the  same 
meanings  as  in  last  section.)  Let  the  ray  be  such  that  OB  and  O'C 
are  parallel.  Since  BC  makes  equal  angles  with  OB  and  O'C,  it 
follows  that  AB  and  CD  make  equal  angles  with  them.  Therefore 
AB  and  CD  are  parallel. 

Let  CB  be  continue^  to  meet  OQ  in  K.  We  have  OB/OB  = 
O'B/O'C,  and  so  the  lengths  of  OE  and  O'K  bear  a  constant  ratio  to 
each  other.  Therefore  K  is  a  fixed  point,  and  is  called  the  centre  of 
the  lens. 

Thus  no  ray  which  passes  through  the  centre  of  the  lens  is 
deviated  from  its  original  direction  by  its  passage  through  the  lens. 


LIGHT  I    REFLECTION,    REFRACTION,   DISPERSION.  217 

It  is  easy  to  find  an  expression  for  the  distance  BQ  in  terms  of 
the  thickness  of  the  lens  and  its  radii  of  curvature.  When  the  lens 
is  very  thin  K  practically  coincides  with  either  Q  or  Q'. 

We  shall  hereafter  assume  that  we  are  dealing  with  thin  lenses 
only. 

The  final  equation  of  §  189  may  be  written  in  the  form  opjop'  = 
I*?  (1— sin2  i}/(i^~  sin2  i),  or,  approximately,  ^  (1— i2)/(/*2— ft).  This 
shows  that,  when  i  is  so  small  that  its  square  may  be  neglected,  p 
coincides  with  p'.  And  it  is  easy  to  prove  that,  when  the  reflecting 


FIG.  111. 

surface  is  spherical  instead  of  plane,  p  and  p1  still  practically  coin- 
cide when  i2  is  small.  (The  formulae  of  §  193  enable  us  to  calculate 
the  positions  of  the  focal  lines.)  Hence  we  conclude  that  a  given, 
sufficiently  small,  pencil  of  rays  will,  after  refraction  through  a  thin 
lens,  be  brought  to  a  focus  at,  or  appear  to  diverge  from,  one 
definite  point ;  and  this  point  must  be  situated  on  the  line  drawn 
from  the  vertex  of  the  incident  pencil  through  the  centre  of  the 
lens. 

We  are  now  in  a  position  to  investigate  the  production  of  images 
by  means  of  lenses. 

195.  Formation  of  Images  by  Lenses. — Let  AB  (Fig.  112)  be  a 
thin  lens,  of  which  C  is  the  centre,  and  CF  is  the  principal  focal 
length.  If  MN  be  an  object  which  is  situated  at  a  distance  from 
the  lens  greater  than  CF,  rays  diverging  from  N  will  be  brought  to 
a  focus  at  a  point  n  such  that  l/CF  =  l/CN  +  l/Cra. 

Similarly  1/CF  =  I/CM  +  I/Cm,  and  so  on. 

Thus  an  inverted  real  image  of  MN  is  formed  at  mn,  and  the  rays 
diverging  from  that  image  may  be  examined  by  an  eye  which  is 


218 


A   MANUAL    OF   PHYSICS. 


situated  at  the  distance  of  about  ten  inches  from  it — that  being  the 
usual  distance  for  most  distinct  vision. 


FIG.  112. 

On  the  other  hand,  if  MN  be  slightly  nearer  (Fig.  113)  to  the  lens 
than  the  principal  focus,  the  lens  is  only  able  to  diminish  the  diver- 
gence of  the  incident  rays,  and  the  erect  and  virtual  image  of  MN 


FIG.  113. 

is  situated  at  a  greater  distance  from  the  image  than  the  object  is. 
For  the  purpose  of  correct  vision  as  regards  an  eye  placed  close 
behind  the  lens,  this  distance  must  be  about  ten  inches. 

The  magnifying  power  of  the  lens  is  the  ratio  of  inn  to  MN,  and 
is  therefore  approximately  equal  to  10//,  where  f  is  the  principal 
focal  length  expressed  in  inches. 

The  object-glass  of  the  ordinary  astronomical  telescope  acts  in  the 
manner  first  described  above.  The  practically  parallel  rays  from  a 
very  distant  object,  MN  (Fig.  114)  are  converged  to  the  principal 
focus  of  the  object-glass,  and  so  an  inverted  image,  mn,  is  formed. 
The  eye-glass  is  placed  at  a  distance  from  mn  which  is  slightly  less 


LIGHT  I     REFLECTION,    REFRACTION,    DISPERSION. 


219 


than  its  principal  focal  length,  and  so  forms  a  magnified  image, 
m'nrt  still  inverted,  at  a  distance  of  ten  inches  from  the  eye. 


N 


The  magnifying  power  of  the  telescope  is  the  ratio  of  the  angle 
which  mm,  subtends  at  the  eye-glass  to  the  angle  which  it  subtends 
at  the  object-glass.  It  is,  therefore,  approximately  equal  to  the 
ratio  of  the  focal  length  of  the  object-glass  to  that  of  the  eye-glass. 

The  arrangement  of  lenses  in  the  compound  microscope  is  essen- 
tially the  same.  The  object  is  placed  at  a  distance  from  the  object- 
glass  which  is  slightly  greater  than  its  principal  focal  length,  and 
so  a  magnified  inverted  image  is  formed,  and  is  further  magnified 
by  the  eye-glass. 

196.  Dispersion  :  Aberration. — In  all  the  preceding  sections  it 
was  assumed  that  we  were  dealing  with  light  of  one  definite  kind  or 
colour  alone.  But  rays  of  light  of  different  colours  are  differently 
refracted  by  any  given  substance  ;  and  we  must,  therefore,  with  this 
in  view,  reconsider  briefly  the  action  of  prisms  and  lenses. 

Let  a  ray  of  white  light,  ab  (Fig.  115),  fall  upon  a  prism,  ABC, 
in  a  direction  perpendicular  to  its  edge.  The  single  ray  of  white 


FIG.  115. 

light  will,  on  entering  the  prism,  be  broken  up  into  a  series  of 
coloured  rays,  fee,  be',  etc.     The  red  rays  are  least  deviated,  and  the 


220  A   MANUAL   OF   PHYSICS. 

blue  or  violet  rays  are  most  deviated,  from  their  original  direction. 
The  rays  intermediate  between  the  red  and  the  violet  are  usually 
broadly  characterised  in  order  as  orange,  yellow,  green,  blue,  and 
indigo. 

Let  be  and  be'  represent  respectively  a  violet,  and  a  red,  ray ;  and 
let  these  lines  meet  the  perpendicular  from  a  on  AB  in  the  points  r 
and  v.  So  long  as  the  square  of  the  angle  of  incidence  is  negligable, 
an  eye  placed  in  the  substance  of  the  prism  will  see,  no  white  point 
a  but,  a  coloured  line,  rv ,  red  at  the  end  nearest  to,  and  violet  at  the 
end  farthest  from,  the  prism. 

After  emergence  from  the  face  BC,  the  two  rays  considered  will 
take  the  directions  cd,  c'd'.  Drop  the  perpendiculars  rr'  and  vv'  on 
BC,  and  let  dc  meet  vv'  at  the  point  v',  while  d'c'  meets  rr'  at  the 
point  rf.  The  angles  which  the  emergent  rays  make  with  BC  being 
small  (which  necessitates  ABC  being  small),  we  see  that  an  eye 
placed  in  the  same  medium  as  the  point  a,  but  on  the  opposite  side 
of  the  prism  from  it,  perceives,  instead  of  a,  a  coloured  line,  r'v'. 

Let  us  suppose  now  that  a  represents  the  section,  by  the  plane  of 
the  paper,  of  a  luminous  white  line,  which  is  parallel  to  the  edge  of 
the  prism.  In  this  case  r'v'  represents  a  coloured  band,  which  is 
called  a  spectrum.  This  separation  of  the  constituents  of  white 
light  by  refraction  is  termed  dispersion.  The  measure  of  the  dis- 
persion produced  by  any  given  substance  is  the  difference  of  the 
refractive  indices  of  that  substance  for  the  extreme  rays  of  the 
visible  spectrum. 

Now  let  P  (Fig.  116)  be  a  luminous  white  point  from  which 
diverging  rays  fall  upon  a  lens  AB.  The  refractive  index  of  the 
substance  of  the  lens  for  violet  rays  being  greater  than  its  refractive 
index  for  red  rays,  the  violet  rays  will  be  brought  to  a  focus  at  a 


FIG.  116. 

point,  P)  which  is  nearer  the  lens  than  the  point  p',  to  which  the  red 
rays  converge.  The  light  in  the  regions  BpA  and  B'jp'A'  is  nearly 
colourless,  but,  on  the  whole,  it  is  somewhat  violet  in  the  former  and 
reddish  in  the  latter.  In  the  region  immediately  outside  BjjA  the  light 
is  red,  while,  in  the  region  immediately  outside  B'p'A',  it  is  violet. 


LIGHT  :     REFLECTION,    REFRACTION,    DISPERSION.  221 

A  lens  is  said  to  produce  aberration  when  it  fails  to  bring  all  rays 
diverging  from  one  point  to  a  focus  at  another  point.  The  aberra- 
tion is  called  chromatic  aberration  when  it  is  due  to  dispersion  :  it 
is  called  spherical  aberration  when  it  is  due  to  the  spherical  form 
of  the  faces  of  the  lens.  The  former  depends  on  the  first  power  of 
•i,  the  angle  of  incidence,  while  the  latter  (§  194)  depends  on  i2.  So 
long  as  the  lens  is  sufficiently  thin,  and  the  incidence  is  sufficiently 
direct,  both  are  negligable. 

197.  Dispersion  :  Achromatism. — It  is  possible  to  get  nearly  rid 
of  dispersion  while  refraction  remains,  and  thus  we  can  obtain  a 
practically  achromatic  lens.  This  result  can  be  obtained,  since 
some  highly  refracting  substances  produce  comparatively  small  dis- 
persion, while  some  substances  of  low  refracting  power  produce 
comparatively  large  dispersion — comparatively,  that  is,  to  their 
refraction. 

Let  us  suppose  that  two  prisms,  of  equal  angles,  but  of  different 
substances,  produce  dispersions  d±,  d9j  respectively,  and  let  their 
refractive  indices  for  the  extreme  red  light  of  the  spectrum  be  plt  /*3 
respectively.  If  now  we  alter  the  angle  of  the  prism,  the  dispersion 
of  which  is  dv  in  the  ratio  d.2/d^  and  form  a  compound  prism  of  the 
two  with  their  edges  turned  in  opposite  directions,  we  shall  have  a 
prism  which  will  not  produce  dispersion  of  the  red  and  violet  rays ; 
but  refraction  will  still  take  place  unless  dj^^de,/^. 

Similarly,  we  may  construct  a  compound  lens  for  the  purpose  of 
avoiding  chromatic  aberration. 

By  no  pair  of  substances  yet  found,  however,  can  we  produce 
complete  achromatism.  If  a  lens  is  completely  achromatic  for  two 
definite  kinds  of  light  it  will  not  be  so  for  any  other  pair.  For  if 
given  dispersion  is  produced  by  one  prism,  between  a  series  of  pairs 
of  definite  kinds  of  light,  equal  dispersion  will  not  be  produced,  by 
any  other  prism,  between  more  than  one  of  these  pairs.  This  is 
known  as  the  Irrationality  of  Dispersion. 

Compound  lenses,  made  up  of  three  constituent  lenses,  can 
produce  closer  approximation  to  achromatism  than  can  a  compound 
lens  made  up  of  only  two. 

193.  Rainbows  :  Halos. — We  are  now  in  a  position  to  discuss  the 
formation  of  rainbows  and  halos. 

Let  AB  (Fig.  117)  represent  a  ray  of  sunlight  which,  falling  on  a 
drop  of  water  at  B,  is  refracted  to  C.  After  reflection  at  C  the  ray 
emerges  at  D.  The  whole  figure  is  symmetrical  about  the  line 
drawn  from  C  through  the  centre  of  the  drop.  The  semi-angle 
between  AB  and  ED  is  obviously  2r  -  i,  i  and  r  being  respectively 
the  angles  of  incidence  and  refraction. 


222 


A   MANUAL    OF    PHYSICS. 


Now  (§  188)  this  is  a  maximum  when  3_sin2  'i  =  4-/*2;  and,  in  the 
case  of  water  and  yellow  light,  /*  =  1-336.'  This  makes  the  semi- 
angle,  corresponding  to  the  maximum  value  of  2r-t,  equal  to 
21°  1'  nearly.  But  the  existence  of  a  maximum  means  that  the 


rays  are  crowded  closely  together  in  the  immediate  neighbourhood 
of  the  maximum  angle,  and  so  an  eye  situated  at  E  will  see  com- 
paratively strong  yellow  light  in  the  direction  ED. 

Now  consider  AB  to  be  a  ray  of  white  light.  This  becomes  dis- 
persed on  refraction,  and  a  blue  ray  (say)  being  more  refracted,  will 
suffer  reflection  at  a  point  C'  such  that  the  angle  %r  —  i  is  smaller 
than  before.  Hence  the  maximum  value  of  the  angle  2r  -  i  becomes 
smaller  as  /*  increases. 

From  E  draw  EP  parallel  to  AB,  and  let  a  very  large  number  of 
drops  be  situated  symmetrically  around  EP  in  the  direction  of  the 
sun's  rays.  (Of  course,  in  the  figure,  the  size  of  the  drop  is 
immensely  exaggerated  relatively  to  other  magnitudes.)  The  eye 
at  E  will  see  a  circle  of  yellow  light,  of  radius  4a±±£;  the  centre  of 
which  is  situated  on  the  line  EP.  Inside  this  a  circle  of  blue  light 
will  be  seen,  and,  outside  it,  a  red  circle  will  appear — the  colours  of 
the  spectrum  succeeding  each  other,  in  order  of  decreasing  refrangi- 
bility,  from  within  outwards.  This  constitutes  the  explanation,  by 
geometrical  optics,  of  the  primary  rainbow.  In  the  actual  bow  the 
colours  are,  of  course,  impure.  For  the  light  proceeding  from  each 
point  of  the  sun's  disc  gives  rise  to  a  separate  bow,  and  the  bow 
which  we  see  results  from  the  superposition  of  all  these  distinct  bows. 

If  the  ray  AB  (Fig.  118)  suffers  two  reflections  in  the  interior  of 
the  drop,  the  emergent  ray  EP  makes  an  angle  with  it  which  is 


LIGHT  :    REFLECTION,    REFRACTION,    DISPERSION. 


223 


double  of  7T/2  —  (3r— i).  But,  §  188,  3r— i  reaches  a  maximum  when 
8  sin-  i  =  9-  /A  Hence  the  angle  BPE  is  a  minimum  when  i  has 
the  value  indicated  by  this  formula.  For  yellow  light,  falling  on  a 
drop  of  water,  the  vertical  angle  is  about  50°  58'. 

A  ray  of  higher  refrangibility  will  be  reflected  at  points  Cf,  D', 
such  that  the  perpendicular  from  0  on  C'D'  will  intersect  AB  at  an 


FIG.  118. 

angle  which  is  larger  than  BPO.  Hence  the  minimum  value  of  the 
angle  7r/2  -  (3r  -  i)  increases  as  //  increases.  And  so,  in  the  secondary 
rainbow,  which  is  due  to  two  such  internal  reflections,  the  colours 
succeed  each  other,  in  order  of  increasing  refrangibility,  from  within 
outwards. 

Also,  since  the  primary  bow  corresponds  to  a  maximum  value  of 
the  angle  which  the  emergent  ray  makes  with  the  incident  ray, 
while  the  secondary  bow  corresponds  to  a  minimum  value  of  this 
angle,  we  see  that  the  space  between  the  bows  is  devoid  of  light  due 
to  rays  which  have  suffered  one  or  two  reflections  inside  the  drops, 
while  there  is  some  such  illumination  in  the  region  inside  the  first 
bow  and  also  in  that  outside  the  second. 

The  bows  which  are  due  to  more  than  two  internal  reflections  are 
too  feeble  to  be  visible. 

Halos  are  due  to  the  refraction  of  light  through  ice-crystals. 
The  rays  are  most  densely  crowded  in  the  directions  of  minimum 
deviation.  The  red  rays  are  least  deviated,  and  therefore  appear 
always  on  the  interior  portions  of  halos.  The  size  of  a  halo  depends 
upon  the  effective  angle  of  the  ice -crystal  to  refraction  through 
which  it  is  due.  Parhelia  and  paraselenes  are  simply  exceptionally 
bright  portions  of  halos. 

Colourless  halos  are  produced  by  the  reflection  of  light  from  the 
plane  surfaces  of  the  crystals. 

199.  The  general  phenomena  of  reflection  and  refraction  receive 
a  ready  explanation  whether  on  the  corpuscular  or  on  the  undulatory 
theory  of  light. 

Let  AB  (Fig.  119)  represent  the  bounding  surface  between  two 
refracting  media.  Let  p  q  r  s  t  represent  the  path  of  a  corpuscle. 


224  A   MANUAL    OF    PHYSICS. 

During  the  rectilinear  motion  from  p  to  q  the  corpuscle  experiences 
no  resultant  attraction  in  any  direction.  When  it  reaches  the 
distance  from  AB  indicated  by  the  line  ab,  the  greater  (say) 
attraction  of  the  medium  on  the  other  side  of  AB  preponderates 
and  the  path  becomes  concave  towards  the  surface  of  separation. 
This  continues  until  a  point,  s,  at  which  the  resultant  attraction 
again  becomes  zero,  is  reached.  The  rest  of  the  path  st  is  therefore 
straight,  and  is  inclined  at  a  less  angle  to  the  normal  to  the  surface 
AB  than  is  the  part  pq. 


a 

A 


a 


FIG.  119. 

Even  if  the  refracting  surface  is  not  actually  plane,  it  is  yet  prac- 
tically plane,  in  all  cases  of  finite  curvature,  so  far  as  the  present 
reasoning  is  concerned,  for  the  portion  qs  of  the  path  of  the  corpuscle 
is  excessively  small. 

Now  (cf.  §  42)  the  square  of  the  resolved  part  of  the  speed  of  the 
corpuscle  along  the  normal  increases  by  a  constant  amount  in  pass- 
ing from  ab  to  a'b',  while  the  resolved  part  along  the  surface  remains 
constant,  and  hence  the  total  speed,  vf,  of  the  particle  in  the  second 
medium  bears  a  constant  ratio  to  its  total  speed,  v,  in  the  first. 
Let,  as  usual,  i  and  r  denote  respectively  the  angles  which  pq  and 
st  make  with  the  normal.  We  have  v  sin  i  =  v'  sin  r,  which  is 
identical  with  sin  i~p  sin  ?*,  where  /*  is  equal  to  v'/v.  But  this  is 
the  known  law  of  refraction.  A 

It  follows  necessarily  that  the  speed  of  a  corpuscle  is  greater  in  a 
dense  medium  than  in  a  rare  one. 

200.  Let  ABC  (Fig.  120)  represent  a  plane  wave-front,  which 
travelling  through  the  air  in  the  direction  indicated  by  the  arrows, 
reaches  the  surface,  ADF,  of  a  refracting  medium.  From  A,  as 
centre,  describe  a  sphere  of  radius,  AF,  such  that  light  will  travel 
over  the  distance  AF  in  the  refracting  medium  in  the  same  time 
that  it  will  describe  the  distance  CF  in  the  air.  Similarly,  from  D 


LIGHT  :     REFLECTION,    REFRACTION,    DISPERSION. 


225 


describe  a  sphere  of  radius,  DQ,  such  that  DQ  bears  to  EF  (DE  is 
parallel  to  ABC)  the  same  ratio  as  the  speed  of  light  in  the  refracting 
medium  bears  to  its  speed  in  air.  It  is  obvious  from  the  construction 
that  all  such  spheres  touch  a  plane,  PQF,  which  is  therefore  the 
wave-front  after  refraction. 


FIG.  120. 

Let  us  denote  the  speeds  of  light  in  the  medium,  and  in  air, 
respectively,  by  the  letters  v',  v.  Let  i  and  r  be  respectively  the 
angles  of  incidence  and  refraction.  Then  CAF  =  i,  AFP  =  r,  and 
CF  =  AF  shit,  AP  =  AF  sin  r.  But  CF/AP  =  v/vr.  Hence  sin  i= 
vjv'  .  sin  r  =  ^  sin  r,  if  \i  =  v\v* ;  and  so  the  known  law  of  refraction 
is  a  consequence  of  the  wave-theory  of  light. 

Observe  that,  in  a  dense  medium  in  which  /*  is  larger  than  unity, 
v'  is  necessarily  less  than  v.  Hence,  on  this  theory,  the  speed  of 
light  must  be  less  in  a  dense  medium  than  it  is  in  a  rare  one.  This 
conclusion  is  in  direct  opposition  to  that  derived  from  the  principles 
of  the  corpuscular  theory ;  and  so  we  are  furnished  with  a  crucial 
test  between  the  two  theories.  The  result  of  experiment  is  (§  178) 
entirely  in  favour  of  the  undulatory  theory.  Consequently,  we  shall 
hereafter  deal  with  the  results  of  this  theory  alone. 

It  is  easy  to  deduce  from  the  above  theory  the  fact  that  the  time 
taken  by  light  to  travel  from  a  given  point  in  one  medium  to  a  given 


A. 


FIG.  12L 

point  in  another  is  a  minimum.  Let  PAQ  (Fig.  121)  be  the  actual 
path  of  light  proceeding  from  P  to  Q,  and  let  PBQ  be  a  very  near 
path.  Draw  AC  and  BD  perpendicular  to  PB  and  AQ  respectively* 

15 


226  A   MANUAL    OF   PHYSICS. 

Then  CB/AD  =  sin  i/sin  r  =  v/v'.  That  is,  CB/v  =  AD/^',  or  AD  and 
CB  are  described  in  equal  times.  Therefore,  and  since  PA  =  PC, 
DQ  =  BQ,  the  paths  PAQ  and  PBQ  are  described  in  equal  times. 
But,  as  B  moves  away  uniformly  from  A,  QA— QB  increases  at  a 
diminishing  rate,  while  PB  — PA  increases  at  an  increasing  rate. 
Hence  the  time  of  description  of  PAQ  is  a  minimum. 

This  law  of  least  time  was  first  given  by  Fermat.  It  is  true  in 
the  case  of  reflection  also  (see  §  182).  The  corpuscular  theory  gives 
PAv+AQ-y'  =  a  minimum.  This  sum  is  termed  the  action. 

201.  Hamilton's  Characteristic  Function. — In  the  earlier  part 
of  the  present  century  Sir  W.  R.  Hamilton  introduced  a  general 
method  by  which  all  optical  problems  may  be  solved  by  a  single 
process.  The  following  extracts  from  an  '  Account  of  a  Theory  of 
Systems  of  Kays,'  written  by  Hamilton  himself,  and  published  in 
his  Life  will  indicate  the  nature  of  his  work. 

'  A  Bay  in  optics  is  to  be  considered  here  as  a  straight  or  bent  or 
curved  line,  along  which  light  is  propagated,  and  a  System  of  Rays 
as  a  collection  or  aggregate  of  such  lines,  connected  by  some  common 
bond,  some  similarity  of  origin  or  production,  in  short,  some  optical 
unity.  Thus  the  rays  which  diverge  from  a  luminous  point  compose 
one  optical  system,  and,  after  they  have  been  reflected  at  a  mirror, 
they  compose  another.  To  investigate  the  geometrical  relations 
of  the  rays  of  a  system  of  which  we  know  (as  in  these  simple  cases) 
the  optical  origin  and  history,  to  inquire  how  they  are  disposed 
among  themselves,  how  they  diverge  or  converge,  or  are  parallel, 
what  surfaces  or  curves  they  touch  or  cut,  and  at  what  angles  of 
section,  how  they  can  be  combined  in  partial  pencils,  and  how  each 
ray  in  particular  can  be  determined  and  distinguished  from  every 
other,  is  to  study  that  System  of  Rays.  And  to  generalize  this 
study  of  one  system  so  as  to  become  able  to  pass,  without  change  of 
plan,  to  the  study  o£  other  systems,  to  assign  general  rules  and  a 
general  method  whereby  these  separate  optical  arrangements  may 
be  connected  and  harmonized  together  is  to  form  a  Theory  of 
Systems  of  Rays.  Finally,  to  do  this  in  such  a  manner  as  to  make 
available  the  powers  of  the  modern  mathesis,  replacing  figures  by 
functions  and  diagrams  by  formulae,  is  to  construct  an  Algebraical 
Theory  of  such  Systems,  or  an  Application  of  Algebra  to 
Optics 

'  The  method  employed  in  that  treatise  (Malus's  Traite 
d'Optique)  may  be  thus  described:  The  direction  of  a  straight 
ray  of  any  final  optical  system  being  considered  as  dependent  on  the 
position  of  some  assigned  point  upon  that  ray,  according  to  some 
law  which  characterizes  the  particular  system  and  distinguishes  it 


LIGHT:   REFLECTION,  REFRACTION,  DISPERSION.  227 

from  others  ;  this  law  may  be  algebraically  expressed  by  assigning 
three  expressions  for  the  three  co-ordinates  of  some  other  point  of 
the  ray,  as  functions  of  the  three  co-ordinates  of  the  point  proposed. 
Malus  accordingly  introduces  general  symbols  denoting  three  such 
functions  (or,  at  least,  three  functions  equivalent  to  these),  and  pro- 
ceeds to  draw  several  important  general  conclusions,  by  very  com- 
plicated but  yet  symmetrical  calculations,  many  of  which  conclusions, 
along  with  many  others,  were  also  obtained  afterwards  by  myself, 
when,  by  a  method  nearly  similar,  without  knowing  what  Malus 
had  done,  I  began  my  own  attempts  to  apply  algebra  to  optics. 
But  my  researches  soon  conducted  me  to  substitute,  for  this  method 
of  Malus,  a  very  different,  and  (as  I  conceive  I  have  proved)  a  much 
more  appropriate  one,  for  the  study  of  optical  systems ;  by  which, 
instead  of  employing  the  three  functions  above  mentioned,  or  at 
least  their  two  ratios,  it  becomes  sufficient  to  employ  one  function, 
which  I  call  characteristic  or  principal.  And  thus,  whereas  he 
made  his  deductions  by  setting  out  with  the  two  equations  of  a  ray, 
I,  on  the  other  hand,  establish  and  employ  the  one  equation  of  a 
system. 

'The  function  which  I  have  introduced  for  this  purpose,  and 
made  the  basis  of  my  method  of  deduction  in  mathematical  optics, 
had,  in  another  connection,  presented  itself  to  former  writers  as 
expressing  the  result  of  a  very  high  and  extensive  induction  in  that 
science.  This  known  result  is  usually  called  the  law  of  least  action, 
but  sometimes  also  the  principle  of  least  time,  and  includes  all  that 
has  hitherto  been  discovered  respecting  the  rules  which  determine 
the  forms  and  positions  of  the  lines  along  which  light  is  propagated, 
and  the  changes  of  direction  of  those  lines  produced  by  reflection  or 
refraction,  ordinary  or  extraordinary.  A  certain  quantity  which  in 
one  physical  theory  is  the  action,  and  in  another  the  time,  expended 
by  light  in  going  from  any  first  to  any  second  point,  is  found  to  be 
less  than  if  the  light  had  gone  in  any  other  than  its  actual  path,  or 
at  least  to  have  what  is  technically  called  its  variation  null,  the 
extremities  of  the  path  being  unvaried.  The  mathematical  novelty 
of  my  method  consists  in  considering  this  quantity  as  a  function  of 
the  co-ordinates  of  these  extremities,  which  varies  when  they  vary, 
according  to  a  law  which  I  have  called  the  law  of  varying  action  ; 
and  in  reducing  all  researches  respecting  optical  systems  of  rays  to 
the  study  of  this  single  function  :  a  reduction  which  presents 
mathematical  optics  under  an  entirely  novel  view,  and  one  ana- 
logous (as  it  appears  to  me)  to  the  aspect  under  which  Descartes 
presented  the  application  of  algebra  to  geometry.' 

15—2 


CHAPTER  XVII. 

RADIATION   AND   ABSORPTION  :    SPECTRUM   ANALYSIS.      ANOMALOUS  DIS- 
PERSION.     FLUORESCENCE. 

202.  WE  have  already  discussed  the  reflection  (including  scattering) 
and  refraction  of  light  at  the  common  surface  of  two  media.  We 
have  now  to  consider  the  absorption  of  light  hi  its  passage  through 
material  media,  together  with  other  associated  phenomena. 

According  to  the  undulatory  theory  (which  we  now  assume  to  be 
true,  and  of  the  truth  of  which  we  shall  receive  additional  evidence 
as  we  proceed)  light  consists  of  waves  propagated  through  a  medium 
(called  the  ether)  which  fills  space. 

The  particles  of  a  body  which  is  emitting  radiation  must  therefore 
be  hi  rapid  vibratory  motion,  and  must  communicate  their  motion 
to  the  ether.  The  parts  of  the  body,  the  vibrations  of  which  are 
communicated  to  the  ether,  may  be  the  molecules,  or  the  constituent 
parts  of  the  molecules,  or  even  the  atoms. 

When  a  bell  is  struck  violently  and  frequently  the  resulting  sound 
is  extremely  complex  and  consists  of  notes,  of  various  pitches,  which 
may  differ  greatly  from  each  other  in  intensity.  The  more  violent 
the  blows  upon  the  bell  become,  and  the  more  rapidly  they  are 
made,  the  more  complex  will  the  clang  be.  New  forced  vibrations 
appear,  and  the  intensity  of  each  of  those  previously  existing  is 
increased.  It  is  only  when  the  blows  are  excessively  feeble  and 
unfrequent  that  the  fundamental  tone  is  heard  by  itself. 

Now  the  molecules  of  a  solid  body,  of  high  temperature,  are  con- 
stantly colliding ;  and  the  vibrations  induced  in  a  molecule  by  one 
collision  do  not  die  out  before  another  is  sustained.  Hence  the 
radiation  given  off  by  such  a  body  consists  of  vibrations  of  many 
periods ;  and,  as  the  temperature  of  the  body  becomes  higher,  vibra- 
tions of  shorter  and  shorter  period  make  their  appearance,  and  the 
intensity  of  all  previously  existing  vibrations  becomes  greater. 
Thus,  if  we  examine  the  spectrum  of  a  body  the  temperature  of 
which  is  gradually  raised,  we  may  at  first  perceive  no  luminous 


RADIATION    AND    ABSORPTION.  229 

radiation  at  all ;  but,  as  the  temperature  rises,  first  red  light  will 
appear,  then  yellow,  and  so  on,  until  a  complete  continuous  spectrum 
is  seen,  and  the  luminosity  of  every  part  gradually  increases  as  the 
temperature  rises. 

In  the  case  of  an  ordinary  gas,  however,  the  molecules  usually 
are  sufficiently  free  from  collisions  to  allow  of  them  vibrating  in 
their  own  proper  modes.  Hence  radiations  of  definite  periods  only 
will  be  emitted ;  and  thus  the  spectrum  of  a  gas  is  discontinuous,! 
and  consists  of  bright  lines.  It  varies  with  the  temperature  and( 
pressure.  As  the  pressure  is  increased  the  lines  broaden  out,  and 
the  spectrum  gradually  becomes  continuous,  like  that  of  a  liquid 
body  or  of  a  solid. 

We  already  know  (§  173)  that  a  body  which  has  a  definite  period 
of  vibration,  and  which  is  at  rest,  may  be  set  in  vibration  by  the 
communication  to  it,  through  an  intervening  medium,  of  vibrations, 
of  its  own  proper  period,  which  are  emitted  by  another  body. 
Hence,  if  radiation  travelling  through  the  ether  enters  a  material 
medium  (solid,  liquid,  or  gas),  and  if  the  natural  period  of  oscillation 
of  the  molecules  coincides  with  the  period  of  some  of  the  ethereal 
vibrations,  the  molecules  will  be  set  in  motion,  and  the  energy  of 
the  radiation  will  be  diminished. 

This  is  the  process  which  is  termed  absorption. 

In  many  cases  (if  not  really  in  most  cases)  the  period  of  the 
induced  vibration  is  longer  than  that  of  the  ethereal  vibration 
which  induces  it  (§  208).  In  almost  all  cases  the  absorbed  energy 
is  manifested  by  a  rise  of  temperature. 

203.  Equality  of  Emissivity  and  Absorptive  Power.  The 
Absorptive  Power  of  a  body,  under  given  conditions,  for  any  definite 
radiation,  is  the  fraction  of  the  whole  incident  radiation  of  that 
kind  which  it  absorbs.  Now  a  black  body  is  one  which  absorbs  all 
the  incident  radiation  ;  so  we  might  define  the  absorptive  power  for 
the  given  radiation  as  the  ratio  of  the  amount  of  it  which  the  body 
absorbs  to  the  amount  of  it  which  a  black  body  would  absorb. 

The  Emissivity  of  a  body,  at  a  given  temperature,  for  any  given 
radiation,  is  the  ratio  of  the  quantity  of  that  radiation  which  it 
emits  to  the  quantity  of  it  which  is  emitted  by  a  black  body  under 
the  same  conditions. 

An  extremely  simple  relation  connects  these  quantities:  The 
Emissivity  and  Absorptive  Power  of  a  body,  at  a,  given  tem- 
perature, for  any  radiation,  are  equal. 

The  proof  of  this  law  was  given  by  Stewart  in  1858.  The  law  is 
(see  §  255)  an  extension  of  the  statement,  made  by  Prevost  about  a 
century  ago,  that  the  radiation  emitted  by  a  body  depends  solely 


230  A   MANUAL   OF   PHYSICS.      ' 

upon  the  nature  of  the  body,  and  upon  its  temperature  ;  and  various 
experimental  illustrations  of  it  were  known  long  before  Stewart's 
proof  was  given.  Brewster  had  shown  that  definite  portions  of  the 
sun's  light  are  absorbed  in  its  passage  through  the  earth's  atmo- 
sphere. Foucault  had  pointed  out  that,  while  the  electric  arc  emits 
(more  freely  than  its  other  radiations)  yellow  light  of  two  definite 
refrangibilities,  the  light  from  one  carbon  pole  is  robbed  of  these  two 
kinds  of  radiation  when  it  passes  through  the  arc.  Stokes  also  had 
explained  this  by  the  analogous  properties  of  sounding  bodies  (§  173). 

It  is  known,  as  an  experimental  result,  that  a  number  of  bodies, 
at  different  temperatures,  placed  inside  an  enclosure  which  neither 
allows  radiation  to  pass  outwards  from  within  it  nor  inwards  from 
without  it,  will  ultimately  arrive  at,  and  maintain,  one  common 
temperature.  But  this  could  not  result  unless  each  body  emitted 
radiation  at  precisely  the  same  rate  as  that  at  which  it  absorbed  the 
radiation.  This  proves  the  law  so  far  as  radiation  as  a  whole  is 
concerned.  [No  such  enclosure  as  has  been  postulated  exists  in 
nature  ;  but  a  polished  reflecting  surface  of  silver  would  form  a 
sufficiently  close  experimental  approximation ;  and,  the  more  nearly 
the  condition  is  satisfied,  the  more  nearly  does  the  result  hold.] 

The  radiation  inside  the  enclosure  must  be  that  of  a  black  body 
at  the  same  temperature,  for  any  one  of  the  bodies  might  be  a  black 
one.  Let  us  suppose  that  one  of  the  bodies  absorbs  one  definite 
radiation  only,  and  allows  all  others  to  pass  freely  through  it. 
(Solutions  of  didymium  salts  approximately  possess  this  property.) 
This  body  must  emit  the  same  kind  of  radiation  as  it  absorbs,  and 
that  to  precisely  the  same  extent ;  otherwise  its  temperature 
would  vary.  This  proves  the  law  as  stated  for  any  definite 
radiation. 

Many  experimental  illustrations  of  the  truth  of  the  law  were 
given  by  Stewart.  Thus,  a  piece  of  red  glass,  held  in  front  of  a 
fire,  appears  red  because  it  absorbs  the  green  and  blue  rays.  If 
placed  in  the  fire  it  becomes  colourless  when  its  temperature 
becomes  equal  to  that  of  the  fire — for  it  then  still  allows  the  red 
rays  to  pass  through  it,  and,  in  addition,  itself  emits  the  rays  which 
it  absorbed.  If  taken  out  of  the  fire  and  held  in  a  dark  room  it 
emits  bluish-green  light — precisely  that  which  it  absorbed. 

Again,  Stewart,  and  also  Kirchoff  (who  arrived  at  the  results 
under  consideration  independently  of,  though  somewhat  later  than, 
Stewart),  showed  that  a  plate  of  tourmaline,  cut  parallel  to  the- 
axis  of  the  crystal,  emits,  when  heated,  light  which  is  polarised 
(Chap.  XIX.)  perpendicularly  to  that  which  it  allows  to  pass,  that 
is,  it  emits  the  rays  which  it  absorbs  when  cold. 


RADIATION    AND    ABSORPTION.  231 

204.  Spectrum  Analysis.  —  In  order  to  examine  the  luminous 
radiation  emitted  by  a  given  body,  we  may  place  in  front  of  the  body 
a  narrow  vertical  slit,  which  is  situated  at  the  principal  focus  of  a 
convex  lens.  The  light  diverging  from  the  slit  is  thus  condensed 
into  a  parallel  beam  which  is  passed  through  a  prism  (usually  of 
dense  glass)  and  so  gives  rise  to  a  spectrum.  This  spectrum  is 
magnified  by  means  of  a  telescope.  Such  an  arrangement  essen- 
tially constitutes  the  instrument  called  a  spectroscope  (or  spectro- 
meter, if  a  graduated  circle  and  vernier  are  attached  for  the  purpose 
of  determining  the  angular  positions  of  the  telescope  when  it  is 
directed  towards  different  parts  of  the  spectrum). 

Let  us  suppose  that  we  are  examining  the  light  emitted  from  a 
highly-heated  lime-ball,  and  that  this  light,  before  falling  on  the  slit, 
passes  through  a  Bunsen.  flame  in  which  metallic  sodium  is  being 
vaporised.  The  lime-ball  alone  would  give  a  continuous  spectrum. 
The  sodium-tinged  flame  alone  would  give  a  discontinuous  spectrum 
consisting  of  two  bright  yellow  or  orange  lines  situated  close  together. 
The  spectrum  actually  observed  is  continuous,  but  has  two  bright  lines 
in  the  same  position  as  those  in  the  spectrum  of  the  sodium  flame. 

If  we  vaporise  the  sodium  in  the  flame  of  a  spirit  lamp  instead  of 
in  a  Bunsen  flame,  everything  else  remaining  the  same,  a  con- 
tinuous spectrum,  crossed  by  two  dark  lines  in  the  positions  of  the 
former  bright  ones,  will  be  seen.  This  was  pointed  out  and  explained 
by  Kirchoff. 

The  cause  lies  in  the  difference  of  the  temperatures  of  the  Bunsen 
flame  and  the  flame  of  the  spirit  lamp.  A  line  will  appear  bright, 
or  dark,  according  as  the  intensity  of  the  radiation  of  that  particular 
kind  from  a  black  body  at  the  temperature  of  the  flame  exceeds, 
or  falls  short  of,  the  intensity  of  the  radiation  of  that  kind  which  is 
emitted  by  the  source.  For,  if  R  be  the  intensity  of  the  given 
radiation  as  emitted  from  the  source,  while  B'=^?R  is  the  intensity 
of  the  light  of  that  kind  emitted  from  a  black  body  at  the  tem- 
perature of  the  flame,  and  p  is  the  radiating  power  (or  absorptive 
power)  of  the  flame  for  that  radiation,  the  intensity  of  the  given 
kind  of  light  which  reaches  the  eye  is  R  —  joR+p^>R  =  E[l-f-,o(_p--l)]  . 
This  quantity  exceeds,  or  falls  short  of,  R,  according  as  p  is  greater, 
or  less  than,  unity.  If  the  source  were  a  black  body,  p  could  not 
exceed  unity  unless  the  temperature  of  the  flame  were  greater  than 
that  of  the  source ;  but,  the  source  not  being  a  black  body,  p  may 
exceed  unity,  although  the  temperature  of  the  flame  is  below  that  of 
the  source,  i.e.,  bright  lines  may  be  visible.  If,  however,  the  differ- 
ence of  temperature  of  the  source  and  flame  be  sufficiently  great,  the 
lines  will  appear  dark. 


232  A   MANUAL    OF   PHYSICS. 

The  spectrum  of  sunlight  was  found  by  Wollaston  and  (later) 
Fraunhofer  to  exhibit  a  number  of  persistent  dark  bands.  In 
accordance  with  the  above  principles  we  conclude  that  these  lines 
are  due  to  absorption.  Some  of  them  can  be  shown  to  be  due  to 
absorption  in  the  earth's  atmosphere,  but  the  great  majority  are 
produced  by  absorption  in  the  (comparatively)  cold  vapours  sur- 
rounding the  hot  body  of  the  sun.  Sufficient  matter  to  produce  such 
absorption  does  not  exist  in  the  space  between  the  earth  and  the  sun. 

Now  we  can  experimentally  determine  the  kinds  of  radiation 
emitted  by  the  hot  vapours  of  the  various  elementary  substances. 
And  if  it  is  found  that  any  of  these  radiations  are  absent  from  the 
spectrum  of  sunlight  it  is  to  be  inferred  that  the  vapours  of  these 
substances  (provided  the  cause  is  not  terrestrial)  are  present  in  the 
regions  immediately  surrounding  the  sun.  In  this  way  it  is  found 
that  a. great  many  substances  existing  on  the  earth's  surface  are 


FIG.  122. 

present  in  the  sun  in  the  form  of  vapour.  The  lines  A,  B  (Fig.  122) 
are  due  to  oxygen,  but  have  been  shown  to  be  caused  by  absorption 
in  the  earth's  atmosphere.  The  lines  C  and  F  are  due  to  hydrogen. 
The  (double)  line  D  is  caused  by  sodium  vapour.  The  (triple)  line  b 
is  produced  by  the  vapour  of  magnesium.  Some  hundreds  of  lines 
are  caused  by  the  presence  of  the  vapour  of  iron. 

In  the  same  way  the  light  emanating  from  any  star,  comet,  or 
nebula,  etc.,  may  be  examined,  and  the  chemical  constitution  of  the 
luminous  body  inferred.  The  various  stars  may  be  classified  into 
several  groups  according  to  the  nature  of  their  spectra,  and  this 
classification  indicates  approximately  their  relative  age.  Generally 
speaking,  the  more  recent  stars  have  bright-line  spectra,  while  the 
older  stars  exhibit  continuous  spectra  crossed  by  numerous  dark 
lines.  The  spectra  of  nearly  extinct  stars,  however,  resemble  those 
of  recent  stars  to  a  considerable  extent. 

If  the  slit  of  the  spectrometer  is  wide  the  various  coloured  images 
overlap  and  produce  an  impure  spectrum  ;  and,  under  the  same  con- 
dition, a  bright  light  broadens  out  and  becomes  indistinct.  But, 
however  narrow  the  slit  may  be,  a  line — even  if  due  to  light  of  one 
definite  refrangibility  alone — has  always  some  finite  breadth.  The 


RADIATION    AND   ABSORPTION.  233 

reason  is  that  the  radiating  molecules  are  in  violent  motion  —  some 
moving  towards,  others  moving  from,  the  observer. 

If  n  be  the  number  of  vibrations  produced  in  the  ether  per  second, 
and,  if  the  molecule  emitting  the  light  be  at  rest  relatively  to  the 
observer,  the  wave-length,  X,  of  the  disturbance  is  given  by  the 
equation  V=nX,  where  V  is  the  speed  of  light.  But,  if  the  molecule 
be  moving,  relatively  to  the  observer,  with  speed  ±v,  the  wave- 
length will  be  given  by  the  equation  V  ±.v  =  n\'  ;  and  the  apparent 
change  of  wave-length  is 


Hence,  the  molecules  of  a  luminous  body  having  all  possible  speeds 
included  between  the  limits  +  v  and  -v,  a  bright  line  in  the  spec- 
trum of  its  light  will  possess  finite  breadth,  even  when  it  corresponds 
to  one  definite  kind  of  radiation  alone. 

This  principle  has  been  applied  to  determine  the  rate  of  rotation 
of  the  sun  on  its  axis,  the  speed  of  projection  of  gases  in  a  solar 
eruption,  and  the  rate  of  motion  of  stars  to  or  from  the  earth. 

205.  Law  of  Absorption  :  Body  Colour.  Dicliroism.  —  Let  R 
be  the  amount  of  radiation  of  some  definite  kind  which  falls  upon 
an  absorbing  medium.  Let  p  (called  the  absorption  co-efficient)  be 
the  percentage  of  this  radiation,  which  is  stopped  by  a  plate  of  the 
medium  of  unit  thickness.  The  quantity  which  passes  through  the 
given  plate  is  therefore  R(l  -p).  A  second  plate  of  the  substance, 
also  of  unit  thickness,  will  stop  the  fraction,  p  of  this  quantity  ;  so 
that  the  amount,  E(l  —  p)2,  passes  through  a  plate  the  thickness  of 
which  is  two  units.  And,  generally,  the  quantity  which  passes 
through  a  plate,  the  thickness  of  which  is  n  units,  is  E(l  -  p)".  This 
practically  vanishes,  however  small  p  may  be  (provided  only  that  it 
is  finite),  when  n  is  sufficiently  great.  Conversely,  the  amount  of 
radiation,  of  the  given  kind,  from  a  sufficient  thickness  of  such  a 
substance,  is  equal  to  that  of  a  black  body  at  the  same  temperature. 
[The  expression  R(l-p)"is  only  true  on  the  assumption  that  p  is 
constant  for  all  radiations  considered.  If  it  is  not  so,  we  must  write 
2  .  E(l-p)"  instead.] 

A  substance  which  absorbs  (say)  red  light,  will  appear  bluish- 
green  when  viewed  by  transmitted  light.  And,  the  greater  the 
thickness  of  the  substance  through  which  the  light  passes,  the 
denser  will  be  the  apparent  colour,  until,  finally,  practically  no 
light  can  pass. 

Hence  a  substance  into  which  light  penetrates  for  a  short  distance 
and  is  then  reflected  out,  will  appear  to  be  coloured,  provided  that 
selective  absorption  takes  place,  and  its  colour  will  be  the  same  as 


234 


A    MANUAL    OF    PHYSIC^. 


that  of  the  light  which  it  transmits.  This  colour  is  termed  the 
body  colour  of  the  substance. 

For  example,  a  mixture  of  blue  and  yellow  pigments  appears  to 
be  green  because,  if  white  light  falls  upon  it,  the  particles  of  the 
blue  pigment  absorb  the  rays  of  small  refrangibility,  while  the  par- 
ticles of  the  yellow  pigment  absorb  the  rays  of  large  refrangibility. 
The  green  rays  alone  are  partially  reflected  by  both  substances,  and 
so  the  mixture  appears  to  be  green.  (A  mixture  of  blue  and  yellow 
lights  is  of  a  purplish  colour.  This  may  be  seen  by  rotating  rapidly 
a  disc,  divided  into  sectors,  some  of  which  are  coloured  yellow,  and 
some  blue.) 

Now  suppose  that  some  substance  absorbs  (say)  red  light  and 
green  light,  and  let  the  coefficient  of  absorption  for  red  light  be 
much  greater  than  the  coefficient  of  absorption  for  green  light. 
Suppose  also  that,  in  the  incident  light,  the  red  rays  are  more 
intense  than  the  green  rays.  It  is  obvious  that,  while  the  intensity 
of  the  red  rays  in  the  transmitted  light  will  exceed  the  intensity  of 
the  green  rays  so  long  as  the  thickness  of  the  substance  is  small, 
after  a  certain  thickness  is  reached,  the  green  light  will  be  trans- 
mitted in  greater  intensity  than  the  red  light.  The  colour  of  such 
a  substance  will  therefore  change  from  a  reddish  hue  to  a  greenish 
hue,  as  seen  by  transmitted  light,  as  its  thickness  increases.  This 
phenomenon  is  known  as  dickroism. 

The  accompanying  diagram  illustrates  these  facts  graphically. 
The  abscissae  of  the  curves  represent  the  thicknesses  of  the  absorbing 
medium ;  the  ordinates  of  one  set  of  curves  represent  the  intensities 
of  the  transmitted  light  of  one  kind,  and  the  ordinates  of  the  other 
set  indicate  the  intensities  of  the  transmitted  light  of  another 
kind,  corresponding  to  the  various  thicknesses.  The  numbers 
accompanying  the  curves  indicate  different  values  of  the  coefficients 
of  absorption.  At  the  point  p  the  high  absorptive  power  (0'7)  of  the 
substance  for  the  originally  more  intense  light  has  diminished  the 
intensity  of  that  light  to  the  same  value  as  that  which  is  exhibited 
by  the  originally  feeble  light,  for  which  the  co-efficient  of  absorption 
is  only  O'l. 

Many  such  bodies  occur  in  nature.  For  example,  glass,  coloured 
with  a  cobalt  salt,  while  it  transmits  blue  light  when  its  thickness  is 
small,  appears  red  by  transmitted  light  when  its  thickness  is  suffi- 
ciently great. 

The  law  of  absorption,  above  stated,  must  be  true  (neglecting 
such  extraneous  effects  as  internal  reflection  or  scattering  of  light) 
so  long  as  the^coefficient  of  absorption  does  not  depend  upon  the 
intensity  of  the  light.  Such  experiments  as  have  been  made  to  test 
this  point  furnish  confirmatory  evidence. 


RADIATION    AND    ABSORPTION.  235 

206.  Surface  Colour:  Metallic  Reflection. — Some  substances 
reflect  from  their  surface  certain  rays  only;  thus  gold  reflects 
yellowish  rays,  and  copper  reflects  reddish  rays.  The  colour  pro- 
duced by  this  '  metallic  reflection '  is  called  surface  colour. 

The  light  which  is  transmitted  by  a  thin  film  of  such  substances 
is  complementary  to  that  which  is  reflected,  that  is,  the  transmitted 
light  and  the  reflected  light  together  make  up  a  light  of  the  same 


FIG.  123. 

composition  as  that  which  was  incident  upon  the  surface.  The 
reflected  light  cannot  be  plane -polarised  at  any  angle  of  incidence 
(Chap.  XIX.). 

Many  substances,  besides  metals,  exhibit  surface  colour  —  for 
example,  thin  films  of  rose  aniline,  or  of  blue  aniline,  etc.,  appear  of 
different  colours  according  as  they  are  viewed  by  transmitted,  or  by 
reflected,  light.  Such  films  may  be  prepared  by  placing  a  layer  of 


236  A    MANUAL   OF   PHYSICS. 

an  alcoholic  solution  of  the  aniline  on  a  plate  of  glass  and  allowing 
the  alcohol  to  evaporate.  The  colour,  as  seen  by  reflected  light, 
varies  somewhat  with  the  angle  of  incidence. 

The  light  reflected  from  such  films  cannot  be  entirely  polarised  at 
any  angle  of  incidence.  It  consists  of  two  parts — a  part  which  can 
be  plane -polarised  at  a  certain  angle  of  incidence  and  is  identical 
with  the  transmitted  light  (which,  in  fact,  constitutes  the  body 
colour  of  the  substance) — and  a  part  which  cannot  be  plane-polarised, 
and  so  resembles  the  surface  colour  of  metals.  The  polarisable  part 
may  be  got  rid  of  by  suitable  means,  so  that  the  remaining  part  may 
be  examined  alone.  In  the  case  of  permanganate  of  potash,  Stokes 
found  that  the  surface  colour  seemed  to  be  due  to  precisely  those 
rays  which  were  absent  from  the  transmitted  light,  or,  which  is  the 
same  thing,  the  body  colour.  Hence,  the  colour  of  the  light  trans- 
mitted through  this  substance  is  due  only  to  a  very  slight  extent,  if 
at  all,  to  absorption.  The  spectrum  of  the  transmitted  light  has 
five  dark  bands  in  the  green  part ;  the  reflected  light  is  green,  and 
the  spectrum  of  the  surface-colour  portion  of  it  consists  of  five  bright 
bands,  which  correspond  to  the  dark  bands  in  the  spectrum  of  the 
transmitted  light. 

207.  Anomalous  Dispersion.  —  In  close  association  with  the 
existence  of  dark  absorption  bands  appears  the  phenomenon  of 
anomalous,  or  abnormal  dispersion. 

In  general,  the  rays  of  greater  wave-length  suffer  refraction,  on 
passage  through  a  prism,  to  a  smaller  extent  than  the  rays  of 
shorter  wave-length.  But,  in  many  substances,  this  rule  does  not 
hold.  Such  media  are  said  to  possess  the  property  of  anomalous 
dispersion. 

Fox  Talbot  was  the  first  to  observe  the  phenomenon,  but  he  did 
not  publish  his  observations  for  about  thirty  years.  In  the  mean- 
time Le  Koux  has  observed  that  iodine  vapour  refracted  red  light 
more  than  it  refracted  blue  light. 

Christiansen,  Kundt,  and  others  have  widely  extended  our  know- 
ledge of  such  substances.  Kundt  has  shown  that  the  property  of 
anomalous  dispersion  is  possessed  by  all  substances  which  exhibit 
surface-colour. 

If  a  continuous  spectrum,  such  as  may  be  given  by  a  glass  prism, 
be  examined  through  another  prism  of  a  substance  which  exhibits 
abnormal  dispersion,  the  spectrum  will  no  longer  be  continuous  but 
will  present  one  or  more  dark  bands.  If  the  second  prism  be  now 
turned  so  as  to  have  its  edge  at  right  angles  to  the  edge  of  the  glass 
prism,  the  parts  of  the  continuous  spectrum  will  be  displaced  from 
their  original  positions  to  an  extent  depending  upon  the  refractive 


RADIATION   AND   ABSORPTION.  237 

index  of  the  substance  for  each  kind  of  light.  The  displacement  of 
the  rays,  in  a  part  of  the  spectrum  close  to  a  dark  band,  but  of 
smaller  wave-length  than  the  absorbed  rays,  is  abnormally  small; 
and  the  displacement  of  the  rays  of  slightly  larger  wave-length  than 
those  which  are  absorbed  is  abnormally  great  (Fig.  124). 


FIG.  124. 

The  general  law,  as  given  by  Kundt,  is  that  the  rays  of  slightly 
less  refrangibility  than  the  absorbed  rays  have  their  refrangibility 
abnormally  increased,  while  the  rays  of  slightly  greater  refrangi- 
bility than  the  absorbed  rays  have  their  refrangibility  abnormally 
diminished  on  passage  through  the  absorbing  medium. 

208.  Fluorescence.  —  The  phenomenon  of  fluorescence  is  also 
necessarily  associated  with  the  absorption  of  light. 

Brewster  observed  that  the  path  of  a  beam  of  white  light  through 
a  solution  of  chlorophyll  glows  with  red  light,  and  he  termed  the 
phenomenon  '  internal  dispersion.'  Then  Herschel  noticed  that  the 
surface  of  a  solution  of  sulphate  of  quinine  upon  which  sunlight 
falls  is  of  a  bright  blue  colour.  He  named  this  appearance  *  epi- 
polic  dispersion  ';  but  Brewster  showed  that  the  blue  colour  could  be 
manifested  in  the  interior  of  the  liquid  if  the  beam  of  sunlight  were 
sufficiently  concentrated,  and  so  he  concluded  that  the  phenomenon 
was  of  the  same  kind  as  that  which  he  had  already  observed  in  the 
case  of  chlorophyll  and  fluorspar,  etc. 

Stokes  has  shown  that  a  great  many  ordinary  substances,  such  as 
bone,  white  paper,  etc.,  possess  this  property,  to  which  (avoiding 
any  reference  to  dispersion)  he  gave  the  name  of  fluorescence,  from 
its  being  noticeable  in  fluorspar.  His  method  of  observation  con- 
sisted in  allowing  a  beam  of  light  to  enter  a  darkened  chamber 
through  a  plate  of  blue  cobalt-glass.  This  beam  fell  partly  upon  a 
white  non-fluorescent  body  (white  porcelain),  and  partly  upon  the 
body  under  examination.  The  light  reflected  from  the  two  sub- 
stances was  then  examined  through  a  slit  and  prism.  In  this  way  the 
fluorescent  light  was  compared  with  the  light  which  produced  it. 

Stokes  found  that  the  light  which  was  emitted  by  the  fluorescent 
body  was  always  of  lower  refrangibility  than  the  light  which  pro- 


238  A    MANUAL    OF    PHYSICS. 

duced  it.  The  fluorescence  of  sulphate  of  quinine  is  due  to  the 
extreme  violet  rays  of  the  spectrum,  and  to  invisible  rays  of  still 
higher  refrangibility.  Hence,  by  means  of  such  a  solution,  the 
absence  of  rays  beyond  the  visible  part  of  a  spectrum  may  be  deter- 
mined. For,  if  the  spectrum  be  thrown  on  a  screen  damped  with 
this  solution,  fluorescence  is  produced,  beyond  the  usual  visible  part, 
except  when  rays  of  certain  refrangibilities  may  be  absent.  In  the 
case  of  chlorophyll  the  light  which  produces  the  effect  is  chiefly  in 
the  visible  spectrum. 

The  explanation  of  the  phenomenon  given  by  Stokes  is  that  the 
ethereal  vibrations  are  absorbed  by  the  fluorescent  matter,  which  is 
set  in  vibration,  the  period  of  its  vibration  being  usually  longer  than, 
never  shorter  than,  the  period  of  vibration  of  the  ether.  The 
vibrating  matter  now  reacts  upon  the  ether,  and  sets  up  in  it  vibra- 
tions which  are  generally  longer  still,  but  are  never  shorter  than  those 
induced  in  the  molecules  of  the  matter.  This  explains  the  lowering 
of  refrangibility. 

Dynamical  illustrations  of  such  interaction  can  be  given.  The 
following  is  due  to  Stokes.  Ships  at  rest  on  a  calm  sea  may  be  set 
in  vibration  by  waves  of  definite  period  propagated  from  a  distance. 
The  natural  period  of  oscillation  of  each  ship  will  not  generally 
agree  with  that  of  the  waves.  Any  ship  which  is  thus  set  in  vibra- 
tion will,  by  its  vibrations,  produce  waves  which  spread  outwards 
from  it ;  but  the  period  of  these  waves  will  generally  be  greater 
than  that  of  the  original  waves,  and  can  never  be  less  than  it. 

If  Stoke's's  explanation  be  true  it  is  to  be  expected  that  the  light 
which  gives  rise  to  fluorescence  will  be  absent  from  the  absorption 
spectrum  of  the  substance.  This  is  invariably  the  case. 

Phosphorescence  is  precisely  the  same  phenomenon  as  fluores- 
cence. The  only  difference  which  subsists  between  the  two  is  a 
difference  of  duration.  Phosphorescence  (so-called)  frequently  lasts 
for  hours  after  the  stimulating  radiation  is  removed  ;  fluorescence  is 
maintained  usually  for  only  a  small  fraction  of  a  second  after  the 
light  ceases  to  fall  on  the  substance. 

Becquerel  demonstrated  and  measured  the  finite  time  of  duration 
of  fluorescence  in  many  substances.  His  apparatus  consisted  of  a 
box  with  perforated  revolving  discs  at  either  end.  The  perforations 
were  so  arranged  that  one  end  of  the  box  was  closed,  while  the 
other  was  open.  The  substance  which  was  to  be  tested  was  placed 
inside  the  box,  and,  on  the  discs  (which  had  a  common  axis)  being 
rotated,  an  intermittent  beam  of  light  passed  through  the  substance. 
No  light  could  pass  out  at  the  end  of  the  box  opposite  to  that  at 
which  the  light  entered  unless  the  substance  were  fluorescent. 


RADIATION    AND    ABSORPTION.  239 

But,  that  condition  being  satisfied,  light  could  pass  through  when 
the  speed  of  rotation  of  the  discs  was  sufficiently  great. 

The  duration  of  fluorescence  is  exemplified  in  the  above  dyna- 
mical illustration  by  the  continued  oscillation  of  the  ships  for  some 
time  after  the  cessation  of  the  disturbance  which  originates  it. 

209.  Theories  of  Dispersion. — Cauchy  was  the  first  to  advance  a 
dynamical  theory  of  dispersion.  He  ascribed  it  to  the  coarse-grained- 
ness  of  the  matter  of  which  the  dispersing  substance  is  composed. 
The  great  difficulty  of  this  theory  is  that,  in  order  to  account  for  the 
observed  values  of  the  refractive  indices  of  substances  such  as  glass, 
etc.,  the  number  of  molecules  of  matter  existing  side  by  side  in  the 
length  of  a  wave  of  light  must  be  assumed  to  be  much  smaller  than, 
from  other  considerations  (§  146),  can  possibly  be  admitted.  Sir  W. 
Thomson  has  recently  shown  that  Cauchy's  hypothesis  can  be  so 
modified  as  to  enable  it  to  surmount  this  difficulty. 

This  hypothesis  leads  to  an  expression  for  the  refractive  index, 
H,  of  any  substance  of  the  form 


where  a,  &,  and  c,  etc.,  are  constants,  and  X  is  the  wave-length. 
This  formula  shows  that  the  refractive  index  increases  as  the  wave- 
length diminishes.  Its  results  accord  very  well  with  experimental 
observations  within  the  range  of  the  visible  spectrum,  but  it  does 
not  apply  well  to  the  invisible  rays  at  the  less  refrangible  end  of 
the  spectrum.  The  various  terms  rapidly  diminish  in  numerical 
magnitude. 

Briot  generalised  Cauchy's  investigation  somewhat,  and  deduced 
the  expression 


. 


which  agrees  better  with  experimental  observations  than  the 
former  does,  and  applies  to  a  much  greater  range  of  wave-lengths. 

The  term  x\z  depends  upon  the  direct  action  assumed  to  exist 
between  the  ether  and  matter. 

Modern  theories  (for  example,  that  of  v.  Helmholtz)  have  regard, 
not  so  much  to  space  relations  —  between  wave-length  and  molecular 
distance  —  >as  to  time  relations  —  -between  the  period  of  vibrations  in 
the  ether  and  the  period  of  free  oscillation  of  the  material  molecules. 

V.  Helmholtz  assumes  the  existence  of  a  viscous  resistance  to  the 
motion  of  the  molecules.  When  the  periods  of  the  ethereal  and  the 
molecular  vibrations  are  identical,  or  approximately  identical, 


240  A   MANUAL    OF   PHYSICS. 

absorption  takes  place,  and,  because  of  the  viscosity,  the  vibrational 
energy  takes  the  form  of  heat. 

Thomson's  results  differ  from  those  of  v.  Helmholtz  chiefly 
because  he  purposely  avoids  the  assumption  of  the  existence  of 
viscosity.  He  obtains  the  equation 


when  ft  is  the  refractive  index,  r  is  the  period  of  vibration  of  the 
ether,  and  xi,  x2,  etc.,  are  the  natural  periods  of  oscillation  of  the 
molecules  arranged  in  ascending  order  of  magnitude.  So  long  as  T 
is  considerably  greater  than  xi  and  considerably  less  than  x2,  this 
equation  will  correspond  to  the  case  of  ordinary  refraction.  As  "7" 
approaches  x±  in  value  the  refractive  index  is  abnormally  increased. 
When  r  is  less  than  xit  ^  is  at  first  negative,  but  afterwards  becomes 
positive,  though  abnormally  small,  as  T  still  further  decreases. 
This  explains  the  existence  of  anomalous  dispersion.  Negative 
values  of  /*2,  which  accompany  anomalous  dispersion,  indicate  the 
existence  of  absorption  or  metallic  reflection. 

Thus  the  high  reflecting  power  of  silver  is,  on  this  theory,  due  to 
the  fact  that  each  one  of  all  the  kinds  of  radiation  which  are  observed 
to  be  reflected  from  it  has  a  vibrational  period  which  is  smaller 
than  the  smallest  of  the  natural  periods  of  oscillation  of  the  mole- 
cules of  silver. 

Again,  when  r  has  such  a  value  that  n°  is  positive,  but  is  less 
than  unity,  the  particular  radiation,  of  which  r  is  the  period,  will 
pass  through  the  substance  more  quickly  than  it  passes  through 
air. 

The  energy  of  the  rapid  vibrations  of  the  molecules  is  gradually 
transmuted  into  energy  of  the  slow  vibrations.  This  explains  fluor- 
escence and  the  radiation  of  heat  from  a  body  which  has  absorbed 
light.  The  molecule  may  be  so  constituted  that  the  fluorescence  (or 
phosphorescence)  may  last  for  a  very  long  time. 


CHAPTEE  XVIII. 

INTERFERENCE.      DIFFRACTION. 

210.  Principle  of  Interference. — If  light  consists  of  undulations, 
propagated  through  the  ether,  the  effects  of  which,  at  any  point  of 
the  ether,  are  superposed  in  precisely  the  same  way  as  are  the 
effects  of  separate  simple  harmonic  motions  (§  52),  we  should 
expect  that  conditions  might  occur  under  which  the  resultant 
motion  at  that  point  would  be  null — while,  under  other  conditions, 
the  resultant  motion  might  be  exceptionally  great.  We  already 
know  that,  for  a  similar  reason,  when  waves  are  propagated  along 
the  surface  of  water  from  two  different  sources,  no  resultant  disturb- 
ance of  the  surface  may  exist  at  certain  points.  So  also  sounds 
from  two  different  sources  may  be  totally  unheard  by  an  ear  placed 
at  certain  positions  within  hearing  distance.. 

In  order  to  produce  continuous  interference  at  given  points  it  is 
absolutely  necessary  that  the  waves  diverging  from  two  sources 
should  be  of  precisely  the  same  period,  as  otherwise  the  resultant 
disturbanoe  would  vary  from  a  minimum  to  a  maximum  alternately. 
Thus,  in  the  case  of  sound,  difference  of  period  gives  rise  to  beats 
which  may  be  observed  by  the  ear. 

Now  the  phase  of  the  vibration  emitted  from  one  point  of  a 
name  has  absolutely  no  relation  with  the  phase  of  that  emitted 
from  any  other  point ;  and  hence  we  cannot  expect  observable  inter- 
ference between  rays  coming  from  different  luminous  sources.  Inter- 
ference of  course  does  occur  between  such  rays  constantly,  but,  in 
general,  the  alternations  between  maximum  and  minimum  effects 
will  succeed  each  other  so  rapidly  that  the  eye  can  perceive  no  varia- 
tion of  intensity. 

Therefore  we  conclude  that,  in  order  that  persistent  interference] 
effects  may  be  observable,  the  two  interfering  rays  must  originally1 
have  proceeded  from  a  common  source. 

More  than  two  centuries  ago  Grimaldi  observed  that,  when 
rays  of  light  from  two  sources  overlapped  each  other  and  fell 

16 


242 


A    MANUAL    OF    PHYSICS. 


upon  a  screen,  the  portion  of  the  screen  which  was  illuminated  by 
the  two  rays  appeared  to  be  darker  than  when  it  was  illuminated 
by  one  ray  alone.  He  allowed  sunlight  to  enter  a  darkened 
chamber  through  two  small  apertures  in  the  shutter.  But  these 
apertures  were  illuminated  by  light  coming  from  all  portions  of  the 
sun's  disc,  and  so  the  effect  which  Grimaldi  observed,  to  whatever 
cause  it  may  have  been  due,  could  not  have  been  produced  by  inter- 
ference. Grimaldi,  indeed,  was  not  looking  for  interference  pheno- 
mena— this  was  not  thought  of  until  150  years  later — he  wished  to 
prove  that  light  was  not  material,  since  two  portions  of  light  appar- 
ently destroyed  each  other.  And  this  reasoning  is  practically  con- 
clusive, for  the  conditions  which  would  have  to  be  assumed,  in 
order  to  make  an  explanation  of  these  phenomena  by  the  emission 
theory  possible,  would  be  so  arbitrary  and  artificial  that  no  one 
could  seriously  advance  them. 

211.  Young's  Experiment. — Young  was  the  first  to  observe  true 
interference  effects.  He  admitted  light  through  a  single  small 
aperture  in  a  shutter  behind  which  he  placed  another  shutter 
pierced  by  two  small  openings.  In  this  way  he  obtained  two 
rays  of  light  which  proceeded  originally  from  a  common  source — 
the  single  opening  in  the  first  shutter.  That  portion  of  a  screen 
which  was  illuminated  by  both  rays  was  crossed  by  alternately- 
arranged  dark  and  bright  bands.  Young  observed  that  the  bands 
became  narrower  when  the  distance  between  the  holes  in  the  second 
screen  was  increased.  He  also  noticed  that  the  effect  disappeared 
if  either  opening  were  closed. 

The  wave  theory  affords  a  ready  explanation  of  the  phenomena 
which  Young  observed. 

Let  A,  A'  (Fig.  125)  represent  the  two  openings  in  the  screen,  and 
let  AP,  AT  be  two  rays  which  each  illuminate  the  point  P  of  the 

I 


.XI 


FIG.  125. 

screen  PN.  M  is  the  central  point  of  AA',  and  MN  is  drawn  per- 
pendicular to  AA'  and  PN.  Denote  the  length  of  AM  (or  A'M)  by 
a  and  the  length  of  MN  by  b,  and  let  x  represent  the  distance  PN. 


LIGHT  :    INTERFERENCE,  DIFFRACTION.  243 

The  waves,  which  travel  along  AP  and  A'P,  start  from  A  and  A' 
respectively  in  the  same  phase.  Consequently  the  point  P  will  be 
bright  or  dark  according  as  AP  -  A'P  is  an  even,  or  an  odd,  multiple 
of  half  a  wave-length. 

Now  AP2=(a+o02+&2  and  A'P2  =  (a-o02+&2.  Therefore  AP2- 
A'P2  =  (AP+A'P)  (AP-  A'P)  =  4ax.  But  (a  and  x  being  very  small 
in  comparison  with  b)  AP-f-A'P  is  approximately  equal  to  26,  and 
so  the  condition  gives 

2ax         X 


and  the  point  P  is  bright  or  dark  according  as  n  is  an  even  or  an 
odd  integer. 

This  formula  indicates  that,  to  the  degree  of  approximation  with 
which  we  are  dealing,  the  locus  of  P,  when  b  varies  and  n  is  con* 
stant,  is  the  straight  line  MP.  The  eaact  locus  is  a  hyperbola,  of 
which  A  and  A'  are  the  foci.  This  follows  at  once  from  the  con- 
dition AP  —  A'P=a  constant. 

A  and  A'  may,  of  course,  represent  narrow  luminous  strips  with 
their  length  perpendicular  to  the  plane  of  the  paper.  The  point  P 
then  corresponds  to  a  dark  or  bright  band  also  perpendicular  to  the 
plane  of  the  paper. 

By  measuring  the  quantities  a,  b,  and  x,  and  by  counting  the 
number,  n,  of  the  particular  band  under  observation,  we  can  calculate 
the  value  of  X. 

The  distance  between  the  n'h  and  the  (n+l)'*  band  is  independent 
of  n,  and  is  therefore  constant  when  a,  b,  and  X  are  fixed. 

212.  Fresnel's  Experiment.  —  In  Young's  experiment  the  beams 
of  light  passed  through  apertures  cut  in  a  solid.  Hence  the  observed 
effects  might  have  been  due  to  diffraction  (  §  224)  .  The  result  was  that 
Young's  explanation  was  not  generally  accepted  ;  but  a  modification 
of  his  experiment,  made  by  Fresnel,  completely  settled  the  matter. 

Light,  diverging  from  the  point  fi  (Fig.  126),  is  reflected  from  two 
mirrors,  OB,  OS,  which  are  hinged:  together  at  0,  and  are  inclined  to 
each  other  at  a  very  small  angle.  After  reflection  the  rays  appear  to 
diverge  from  A  and  A',  the  images  of  B  in  OS  and  OB  respectively. 
Hence  A  and  A'  act  as  two  sources  of  :  light,  the  radiation  emitted 
from  each  of  which  is  similar  in  all  respects  to  that  emitted  from  the 
other.  The  light  has  nowhere  passed  through  an  aperture,  so  that 
the  objection  made  to  Young's  form  of  the  experiment  does  not 
apply,  and  yet  the  same  effects  are  observed  to  occur. 

The  points  A',  A,  and  ^f  obviously  lie  on  a  circle,  the  Centre  of 
which  is  at  0  ;  and  the  lines  OB  and  OS  are  respectively  perpendi- 

16—2 


244 


A    MANUAL    OF    PHYSICS, 


cular  to  A'B  and  AB.     Hence  the  angle  A'BA  is  equal  to  the  angle 
of  inclination  of  the  mirrors  =  9  (say) ._    But  A'OA  =  2A'B A  =  20  ;  and 


FIG.  126. 

OM  is  practically  equal  to  OB  =  r  (say).     Therefore,  if  we  denote 
ON  by  r\  the  formula  of  last  section  becomes 

X 


Very  accurate  adjustments  are  necessary  in  order  to  obtain  good 
results  from  this  form  of  the  experiment. 

213.  Lloyd's  Experiment.  —  Lloyd  repeated  the  above  experiment 
with  only  one  mirror.  A  ray  of  light  diverging  from  a  slit,  A' 
(Fig.  127),  is  reflected  in  part,  at  grazing  incidence,  from  a  mirror 


N 


FIG.  127. 


KS.  We  thus  obtain  two  rays  of  light,  one  actually  diverging 
from  A',  and  the  other  apparently  diverging  from  A,  the  image  of  A' 
in  BS  ;  and  these  rays  produce  interference  effects  as  formerly. 

Yet  one  distinct  difference  is  observable.  In  both  forms  of  the 
experiment  previously  described  the  point  N  is  brightly  illuminated, 
for  AN  -  A'N  =  0.  In  Lloyd's  experiment  N  is  dark,  and  the  whole 
system  of  bright  and  dark  bands  is  shifted  by  the  breadth  of  one 
band.  In  explanation  of  this  Lloyd  suggested  that  the  phase  is 
altered  by  180°  in  the  act  of  reflection. 


LIGHT  :    INTERFERENCE,  DIFFRACTION. 


245 


In  all  cases  the  slit  through  which  the  light  passes  should  be 
narrow ;  but  this  is  not  of  so  much  importance  in  the  present  case 
as  in  the  previous  cases.  For  the  slit  A  is  the  inverted  image  of  A', 
and  so  M  is  the  centre  of  all  corresponding  parts  of  A  and  A',  the 
part  of  A  which  is  nearest  to  M  being  the  image  of  the  part  of 
A'  which  is  nearest  to  M,  and  so  on.  Hence  the  effects  of  all  the 
parts  are  strictly  superposed  at  P.  But,  in  the  two  previous  cases, 
since  there  is  no  inversion  of  A'  with  respect  to  A,  the  part  of  A' 
which  is  nearest  to  M  corresponds  to  the  part  of  A  which  is  farthest 
from  M  (M  being  taken  as  the  middle  point  of  the  line  joining  the 
central  parts  of  the  slits),  and  so  on.  Hence  the  systems  of  bands 
due  to  the  light  from  the  various  corresponding  parts  of  the  two 
slits  are  not  exactly  superposed,  and  the  definition  is  in  consequence 
less  accurate. 

214.  Fresnel's  Biprism. — A  second  form  of  the  experiment,  to 
which  also  the  objection  taken  to  Young's  experiment  does  not  apply, 
is  due  to  Fresnel.  ES  (Fig.  128)  is  a  glass  prism  of  very  obtuse  angle. 
It  is  placed  with  its  flat  face  towards  M,  the  source  of  light.  Each 
half  of  the  prism  forms  an  image  of  M,  so  that  the  rays  emerge  from 


the  other  faces  of  the  prism  as  if  they  proceeded  from  points  A  and  A', 
which  are  practically  situated  on  a  straight  line,  through  M,  drawn 
perpendicular  to  the  flat  face  of  the  prism.  If  i^  r1?  are  the  angles 
of  incidence  and  refraction  at  the  flat  face  of  the  prism,  while  i,,  r2,  are 
the  similar  angles  at  the  opposite  face,  the  total  deviation  of  the  ray 
ME,  i.e.,  the  angle  A'EM  is  (§  192)  i^n+i^-r^  These  angles 
being  small,  ii  and  i.2  are  respectively  equal  to  pr^  and  \ir^  \i  being 
the  refractive  index  of  the  substance  of  which  the  prism  is  composed. 
Hence  the  deviation  is  (p  -1)  (r1+r2)  =  (/t—  1)«,  where  a  is  the  acute 
angle  of  the  prism.  This  gives  A'M(  =  AM)  =  &(/*-  l)a  approxi- 
mately, b  being  the  distance  of  M  from  the  biprism,  and  so  the 
formula  giving  the  value  of  x  becomes 

l)ax       \, 
~ 


where  b'  is  the  distance  between  the  prism  and  the  screen. 


246  A   MANUAL   OF   PHYSICS. 

215.  Coloured  Interference  Bands. — In  the  immediately  pre- 
ceding sections  we  have  assumed  the  wave-length  to  be  constant. 
But  the  breadth  between  two  adjacent  bright  or  dark  bands  is  pro- 
portional to  X,  and  so  the  band  situated  at  N  is  the  only  one  which 
is  colourless.  All  other  bands  are  coloured,  the  first  red  band  being 
about  twice  as  far  from  tN  as  the  first  violet  band.  About  a  dozen 
of  these  bands  can  be  fairly  well  distinguished  when  ordinary  white 
light  is  used ;  but  the  succeeding  bands  of  different  colours  are  so 
superposed  that  all  traces  of  interference  effects  practically  disappear, 
and  the  screen  seems  uniformly  illuminated. 

If  the  quantity  a  in  the  formula  of  §  214  were  variable  and  pro- 
portional to  X,  x  would  be  constant  for  all  wave-lengths,  that  is,  the 
bands  would  be  colourless.  This  effect  may  be  attained  by  the  use 
of  a  diffraction  grating  (§  233). 

In  the  biprism  method  the  distance  between  the  points  A  and  A' 
depends  upon  the  wave-length,  and  is  greater  the  shorter  the  wave- 
length is.  The  result  is  that  the  coloured  bands  are  more  widely 
separated  than  they  usually  are. 

The  introduction  of  a  coloured  glass,  which  diminishes  the 
number  of  different  kinds  of  light  in  the  interfering  beams,  produces 
a  very  marked  increase  in  the  number  of  bands  which  are  visible. 
As  many  as  200,000  bands  have  been  counted  when  a  flame,  tinged 
deeply  orange  by  burning  sodium,  was  employed  as  the  source  of  light. 

When  the  difference  between  the  lengths  of  the  paths  travelled  by 
the  two  interfering  rays  is  a  very  large  multiple  of  the  wave-length, 
the  nature  of  the  vibrations  may  have  completely  altered  in  the 
interval  of  time  between  the  setting  out,  from  the  source,  of  the  two 
waves  which  simultaneously  reach  P,  so-  that  no  interference  could 
occur.  But  the  fact  that  no  more  than  200,000  bands  have  ever 
been  counted  does  not  prove  that  no  more  than  200,000  vibrations 
of  the  ether  at  a  given  point  are  sufficiently  nearly  similar  to  pro- 
duce continued  interference,  for  we  can  neither  obtain  absolutely 
monochromatic  light  nor  use  an  infinitely  narrow  slit.  Yet  the 
converse  statement,  that  200,000  successive  vibrations  are  practically 
similar,  is  true. 

216.  Displacement  of  Bands  by  Befracting  Media. — If  a  dense 
medium  be  placed  in  the  path  of  one  of  the  two  interfering  rays, 
the  whole  system  of  bands  will  be  displaced  towards  that  side  of 
MN  on  which  the  medium  is  placed.  For  if  t  be  the  thickness  of  a 
medium  of  refractive  index  //,  which  is  traversed  by  the  ray,  the 
effect  is  the  same  as  if  the  ray  had  traversed  a  thickness,  pi,  of  air. 
Thus  the  effective  length  of  the  path  of  that  ray  is  increased  by  the 
amount  (/i  —  1)  t. 


LIGHT  :    INTERFERENCE,  DIFFRACTION. 


247 


Let  L  (Fig.  129)  represent  the  medium  interposed  in  the  path  of 
the  ray  AT.  The  effective  length  of  A'P  is  increased,  and  so  the 
length  of  AP  must  be  increased.  In  other  words,  PN  must  increase. 

Suppose  now  that  L  is  removed,  and  that  we  shift  A'  back  from 
the  screen  through  the  distance  (^  -  1)^/2  into  the  position  A/.  Let 
also  A  be  moved  towards  the  screen,  through  the  same  distance, 


FIG.  129. 

into  the  position  Ax.  In  this  way  the  effective  length  of  A'N  is 
increased,  relatively  to  that  of  AN,  by  the  amount  (n  —  l)t  ;  and  the 
central  band,  originally  at  N,  will  now  be  found  at  Q,  which  is  such 
that  MQ  is  perpendicular  to  AiA'j.  But  QN/MN  =  AAj/AM  = 
(/*  —  l)£/2a.  Hence  the  displacement  of  the  central  band  (if  we  are 
dealing  with  monochromatic  light)  is 


When  the  light  is  not  monochromatic  the  displacement  of  the  central 
(which  is  then  the  brightest)  band  could  only  be  given  by  this  for- 
mula if  the  refracting  substance  did  not  produce  dispersion,  i.e.,  if 
^  were  independent  of  X.  The  brightest  effect  will  really  be-  pro- 
duced at  a  place  where  the  rate  of  variation  of  QN  with  X  is  a 
minimum,  for,  at  such  a  place,  the  various  adjacent  coloured  bands 
are  most  nearly  superposed. 

By  means  of  the  formula  the  refractive  index  of  the  interposed 
substance  may  be  found  with  extreme  accuracy.  The  method  is 
specially  applicable  to  the  determination  of  the  refractive  indices  of 
gases. 

217.  Interference  Bands  in  Spectra.  —  If  a  beam  of  white  light, 
diverging  from  a  narrow  slit,  be  made  parallel  by  a  suitable  lens, 
and  then  be  refracted  by  a  prism,  the  usual  continuous  spectrum 
will  be  obtained.  But  if  a  plate  of  a  refracting  substance  be  inter- 
posed in  the  path  of  one  half  of  the  beam,  the  spectrum  will  be 


248  A    MANUAL    OF    PHYSICS. 

crossed  by  dark  bands.  The  reason  is  that  one  half  of  the  rays  are 
retarded  relatively  to  the  other  half,  and  so  interference  effects  are 
produced.  Those  rays,  the  relative  retardation  of  which  amounts 
to  a  semi-wave-length,  are  obliterated. 

Various  forms  of  this  experiment  are  described  by  Powell,  Fox, 
Talbot,  Brewster,  and  Stokes. 

218.  Colours  of  Thin  Plates.  Reflected  Ltght.—Thm  films  of 
transparent  substances  are  frequently  observed  to  be  brilliantly 
coloured.  The  colours  vary  with  the  angle  of  incidence  and  with 
the  thickness  of  the  film. 

Familiar  examples  occur  in  the  cases  of  a  soap  bubble,  of  the 
wing  of  the  common  house  fly,  and  of  highly  tempered  steel,  etc. 
In  the  latter  case  the  thin  film  consists  of  an  oxide  formed  on  the 


FIG.  130. 

surface  of  the  steel  at  a  high  temperature.  Very  old  glass  vessels 
frequently  exhibit  these  colours  from  the  partial  splitting  away  of 
thin  films  at  the  surface  of  the  glass. 

The  wave  theory  gives  a  complete  explanation  of  these  pheno- 
mena. 

Let  AB  represent  a  film,  of  (small)  thickness  £,  of  a  substance  the 
refractive  index  of  which  is  //.  A  ray,  ab,  falling  upon  the  upper 
surface  of  the  plate  is  partially  reflected  along  be  and  in  part  is 
refracted  along  bd.  The  refracted  ray  suffers  partial  reflection  in 
the  direction  db'  and  finally  emerges  from  the  substance  in  the 
direction  b'c'  parallel  to  be. 

If  perpendiculars  b'm  and  b'n  be  dropped  from  b'  upon  be  and  bd, 
the  parts  bm,  bn  of  these  paths  intercepted  between  b  and  the  feet 
of  the  perpendiculars  are  described  in  equal  times.  Hence  the 
effective  difference  of  path  described  by  the  two  rays  is  nd-\-db', 
which  is  equal  to  2£  cos  ?•,  where  r  is  the  angle  of  refraction.  And 
this  portion  is  described  in  a  substance  of  refractive  index  ^  so  that 
the  equivalent  path  in  air  is 

2/*£  cos  r. 

It  might,  therefore,  be  expected  that  the  effects  of  the  two  rays 
would  be  mutually  intensified  when  this  quantity  is  an  integral 


LIGHT  :    INTERFERENCE,    DIFFRACTION.  249 

multiple  of  a  wave-length,  that  they  would  mutually  annul  each 
other  when  it  is  an  odd  multiple  of  a  semi-wave-length,  and  that, 
when  the  thickness  of  the  plate  is  much  smaller  than  a  semi-wave- 
length of  violet  light,  all  rays  would  be  intensified,  so  that  white 
light  would  be  reflected. 

But  the  exact;  reverse  of  these  effects  are  observed.  When  the 
thickness  of  the  plate  is  very  small  no  light  is  reflected,  and,  when 
the  quantity  ^\ii  cos  r  is  an  odd  multiple  of  half  a  wave-length,  the 
light  is  strongly  reflected.  These  results  are  precisely  those  which 
would  occur  if,  in  the  acts  of  reflection  at  the  upper  and  under  faces, 
a  difference  of  phase  of  half  a  period  were  introduced. 

The  conditions  under  which  the  two  reflections  take  place  are 
exactly  opposed  to  each  other.     In  the  one  case  the  light  is  passing 
from  a  rarer  into  a  denser  medium  :  in  the  other  it  is  passing  from 
a  denser  into  a  rarer  medium.     Hence,  reasoning  by  analogy  from 
the  effects  of  impact  of  two  elastic  balls  of  different  masses,  Young 
pointed  out  that  the  relative  acceleration  of  phase  which  seems  to  be 
required  ought  to  be  produced.      [The  propagation  of  waves  along  a   ^ 
rope  composed  of  two  parts  of  different  linear  densities,  is  precisely  , 
analogous.     A  wave  propagated  along  the  less  dense  portion  is  in  part 
reflected  from  the  junction  with    a   complete   reversal  of  phase. 
(As  an  extreme  case  imagine  the  rope  to  be  fixed  at  the  junction.  Jj 
This  corresponds  to  infinite  density  of  the  second  part.)     A  wave 
travelling  along  the  more  dense  portion  is  partly  reflected  at  the 
junction  without  change  of  phase.] 

Young  pointed  out  that  if  his  explanation  were  correct  an  entire 
reversal  of  the  effects  should  occur  when  the  reflecting  plate  was 
intermediate  in  density  between  the  media  on  either  side  of  it. 
Further,  he  carried  out  such  an  experiment,  and  found  that  his  pre- 
diction was  verified.  Lloyd's  experiment  (§  213)  furnishes  another 
verification  of  the  correctness  of  Young's  explanation. 

The  effective  difference  of  path,  2ju£  cos  r,  decreases  as  the  angle 
of  incidence  increases,  and  therefore  the  wave-length  of  the  reflected 
light  decreases  as  the  angle  of  incidence  increases.  If  the  refractive 
index  and  the  thickness  of  the  plate  be  sufficiently  large  the  series  of 
colours  may  be  repeated  a  number  of  times,  but,  if  ordinary  white 
light  be  used,  partial  overlapping  will  occur  between  all  the  series 
above  the  second,  for  the  wave-length  of  the  extreme  red  light  of 
the  spectrum  is  approximately  double  of  that  of  the  extreme  violet 
light. 

219.  The  above  explanation  of  the  reflection  of  light  from  thin 
plates  is  not  quite  complete.  The  intensity  of  the  reflected  ray,  be, 
is  always  greater  than  that  of  the  ray  b'c',  and  so  complete 


250  A   MANUAL    OF    PHYSICS. 

annulment  of  light  is  not  accounted  for.  But  complete  annulment 
does  take  place.  A  complete  treatment  of  the  problem  was  given 
by  Poisson,  who  pointed  out  that  all  the  various  rays  which  emerge 
at  V  must  be  taken  into  account.  The  ray  which  enters  at  b  and 
suffers  one  internal  reflection  at  d  before  it  passes  out  of  the  plate  at  b' 


\AA/V 


fed 
FIG.  131. 

has  the  greatest  effect  in  producing  the  final  result ;  but  the  ray  which 
suffers  two  such  internal  reflections  (at  e  and  d)  before  emergence 
also  has  a  considerable  effect.  Similarly,  those  which  have  under- 
gone three,  four,  etc.,  internal  reflections,  have  each  an  appreciable, 
though  rapidly  diminishing,  share  in  the  ultimate  result. 

The  effective  difference  of  path  between  the  ray  which  has 
suffered  n  such  internal  reflections,  and  the  ray  which  is  once  reflected 
externally  at  b  is  ^n\it  cos  r+X/2,  the  semi- wave -length  being  added 
in  order  to  take  account  of  the  acceleration  of  phase  produced  in  the 
act  of  reflection  at  b.  This  being  taken  into  consideration  it  is 
found  that  the  intensity  of  the  light  reflected  from  the  plate  does 
vanish  when  2/i£  cos  r  is  an  even  multiple  of  X/2,  and  that  it  is  a 
maximum  when  2/j.t  cos  r  is  an  odd  multiple  of  X/2. 

220.  Colours   of  Thin  Plates.     Transmitted  Light — The  light 
which  is  transmitted  through  the  plate  is  complementary  to  that 
which  is  reflected  from  it ;  that  is,  the  kinds  of  light  which  are 
absent  from  the  reflected  beam  are  precisely  those  which  are  present 
in  the  transmitted  beam. 

The  intensity  of  the  reflected  light  is  never  equal  to  that  of  the 
incident  light,  and  so  the  intensity  of  the  transmitted  beam  never 
entirely  vanishes.  Also,  since  the  minimum  intensity  of  the 
reflected  light  is  zero,  the  maximum  intensity  of  the  transmitted 
light  is  equal  to  the  intensity  of  the  incident  light. 

221.  Newton's  Rings. — Newton  observed  the  colours  produced  by 
the  interference  of  rays  reflected  from  both  sides  of  a  thin  film  of 
air  enclosed  between  two  pieces  of  glass.     One  of  the  pieces  of  glass 
had  a  plane  surface  ;    the  surface  of  the  other  was  convex  and 
spherical. 


LIGHT  :    INTERFERENCE,  DIFFRACTION.  251 

The  thickness  of  the  film  of  air  at  a  distance,  d,  from  the  point  of 
contact  is  approximately  d2/2B  where  E  is  the  radius  of  the  spherical 


FIG.  132. 

surface.  Hence  the  condition  that  light  of  wave-length  X  shall  be 
intensified  is 

u'd? 
rR-  cos  r 

n  being  any  integer,  and  so  the  point  of  contact  is  surrounded  by  a 
series  of  bright  rings.  In  this  formula,  r  is  the  angle  of  refraction 
from  glass  into  air  and  n'  is  the  reciprocal  of  the  refractive  index  of 
glass.  The  radii  of  successive  bright  rings  are  therefore  given  by 


sec  r  (2w+l)/2  .  X, 

where  /i  is  the  refractive  index  of  the  glass  referred  to  air. 

It  follows  from  this  formula  that  the  replacement  of  the  film  of  I  \ 
air  by  a  denser  substance  would  cause  all  the  rings  to  close  in  some- 1 1 
what  towards  their  common  centre.     This  result  is  proved  by  experi-l  | 
ment,  and  hence  we  get  another  proof  of  the  fact  that  light  travels 
slower  through  a  medium  such  as  water  than  it  does  through  air. 

The  successive  radii  are  proportional  to  the  square  roots  of  the 
natural  numbers — of  the  even  numbers  in  the  case  of  the  dark 
rings,  and  of  the  odd  numbers  in  the  case  of  the  bright  rings,  and 
so  successive  rings  enclose  equal  areas. 

The  radii  also  increase  as  the  wave-length  increases,  and  so  the 
first  red  ring  is  farther  from  the  centre  than  the  first  blue  one  is. 

Lastly,  d  increases  when  the  angle  of  incidence  increases. 

When  the  two  pieces  of  glass  are  pressed  sufficiently  close 
together  a  black  spot  appears  at  the  centre.  The  central  thickness 
is  then  very  small  in  comparison  with  the  wave-length  of  any 
visible  light,  and  so  the  reflected  light  vanishes ;  for  the  effective 
length  of  the  paths  traversed  by  rays  which  emerge  at  a  given 
point  after  internal  reflection  is  practically  the  same  as  that  of  the 
light  which  is  directly  reflected  at  the  same  point  without  entering 
the  thin  film,  and  so  the  two  sets  of  rays  practically  differ  in  phase 
by  half  a  period. 

The  transmitted  light  is  complementary  to  that  which  is  reflected. 
The  central  portion  is  therefore  white. 


252 


A   MANUAL    OF    PHYSICS. 


Theory  indicates  that  if  the  refractive  index  of  the  film  be  inter- 
mediate between  the  indices  of  the  two  transparent  media  which 
bound  it,  the  rings  seen  by  reflection  should  commence  from  a 
white  centre.  Young  verified  this  prediction  by  means  of  a  film  of 
oil  of  sassafras  enclosed  between  a  lens  of  crown  glass  and  a  lens  of 
flint  glass. 

222.  Colours   of  Mixed  Plates. — If   a   bright  object  be  viewed 
through  an  intimate  mixture  of  two  media  of  different  refractive 
indices  (e.g.,  a  mixture  of  oil  and  air  enclosed  between  glass  plates), 
colours  are  observed  to  which  Young  gave  the  name  of  '  colours  of 
mixed  plates.'     The  colours  are  arranged  in  rings  precisely  as  in  the 
case  of  those  seen  by  transmission  of  light  through  a  thin  homo- 
geneous plate,  but  the  whole  system   is  on  a  larger  scale.     The 
phenomenon   is   due  to  the  interference   of   the   rays  which  pass 
through  the  different  media  and  so  suffer  relative  change  of  phase. 

When  the  incident  light  is  oblique  and  a  dark  object  is  placed 
behind  the  plates,  the  system  resembles  that  which  is  ordinarily 
seen  by  reflection,  for  one  of  the  interfering  portions  is  reflected  and 
undergoes  the  usual  acceleration  of  phase. 

223.  Colours  of  Thick  Plates. — Brewster  observed  that,  in  certain 
circumstances,  interference  may  be  produced  by  means  of  plates  the 
thickness  of  which  is  not  small  in  comparison  with  the  wave-length 
of  light. 

AB  and  BC  (Fig.  133)  represent  two  such  plates  of  parallel  glass, 
which  are  precisely  equal  in  thickness,  and  are  inclined  to  each 
other  at  a  small  angle,  «. 

A  pencil  of  light,  Pm,  falls  perpendicularly  upon  the  plate  BC, 
and,  passing  through  it,  is  partly  reflected  from  the  first  surface  of 


FIG.  133. 

AB  and  in  part  is  refracted  into  the  plate  AB.  A  portion  of  the 
refracted  part  is  reflected  at  ra.  If  r  be  the  angle  of  refraction  in 
AB  the  effective  difference  of  path  so  produced  between  the  two 


LIGHT  :    INTERFERENCE,  DIFFRACTION.  258 

portions  of  light  is  Ipt  cos  r  =  2/i£  cos  sin-1(l/^  .  sin  a),  where  /*  is 
the  refractive  index  and  t  is  the  thickness  of  the  glass.  A  similar 
action  occurs  at  the  plate  BC,  and  the  rays  which  were  reflected 
from  the  first  surface  of  AB  sustain,  relatively  to  the  other  rays,  an 
effective  increase  of  path  to  the  amount  Ipt  cos  r'  =  2^£  cos  sin-1 
(l//z  .  sin  2«),  r'  being  the  angle  of  refraction  in  BC.  The  effective 
difference  of  path  of  the  two  rays,  pq,  which  finally  emerge  from 
the  side  of  AB  remote  from  P,  is  therefore 

2ju£(cos  sin"1^  sin  «)  -  cos  sin     (     sin  2a)  ). 

Interference  occurs  when  this  quantity  is  sufficiently  small. 

Jamin  has  applied  this  principle  to  the  construction  of  an 
extremely  sensitive  instrument  for  the  measurement  of  refractive 
indices. 

Newton  observed  interference  effects  when  he  allowed  light  to 
fall  upon  the  surface  of  a  concave  glass  mirror  which  was  silvered 
behind.  The  mirror  was  everywhere  of  uniform  thickness  and  the 
light  was  admitted  through  a  small  opening  in  a  sheet  of  white 
paper — the  opening  being  situated  at  the  centre  of  curvature 
of  the  mirror.  A  few  broad  coloured  rings,  resembling  those  due 
to  light  transmitted  through  a  thin  plate,  were  seen  on  the  paper. 
All  these  rings  were  concentric  with  the  opening  through  which  the 
light  passed. 

The  origin  of  these  colours  is  totally  different  from  that  of  the 
colours  which  Brewster  observed.  The  rings  are  due  to  the  inter- 
ference of  light,  ordinarily  reflected  at  the  silvered  surface  of  the 
mirror  and  then  scattered  by  particles  of  dust  on  the  first  surface, 
with  light,  also  reflected  from  the  silvered  surface,  but  which  had 
been  previously  scattered  (or,  rather,  diffracted}  by  particles  of  dust 
upon  the  first  surface  of  the  mirror. 

When  the  mirror  is  slightly  inclined  the  centre  of  the  coloured 
rings  is  situated  midway  between  the  opening  in  the  paper  and  the 
image  of  it  which  is  formed  upon  the  paper.  This  central  spot  is 
alternately  bright  and  dark  (when  homogeneous  light  is  used),  as 
the  distance  between  the  opening  and  its  image  increases ;  it 
undergoes  a  rapid  variation  of  colour  when  the  incident  light  is 
white. 

224.  Diffraction. — The  principle  by  means  of  which  Huyghens 
explained  the  rectilinear  propagation  of  light  has  already  been 
given  (§  186).  The  following  remarks  of  Stokes  on  this  subject  are 
specially  worthy  of  note. 


254  A   MANUAL    OF    PHYSICS. 

'  When  light  is  incident  on  a  small  aperture  in  a  screen,  the  illu- 
mination at  any  point  in  front  of  the  screen  is  determined,  on  the 
undulatory  theory,  in  the  following  manner.  The  incident  waves 
are  conceived  to  be  broken  up  on  arriving  at  the  aperture  ;  each 
element  of  the  aperture  is  considered  as  the  centre  of  an  elementary 
disturbance,  which  diverges  spherically  in  all  directions,  with  an 
intensity  which  does  not  vary  rapidly  from  one  direction  to  another 
in  the  neighbourhood  of  the  normal  to  the  primary  wave,  and  the 
disturbance  at  any  point  is  found  by  taking  the  aggregate  of  the 
disturbances  due  to  all  the  secondary  waves,  the  phase  of  vibration 
of  each  being  retarded  by  a  quantity  corresponding  to  the  distance 
from  its  centre  to  the  point  where  the  disturbance  is  sought.  The 
square  of  the  co-efficient  of  vibration  is  then  taken  as  a  measure  of 
the  intensity  of  illumination.  Let  us  consider  for  a  moment  the 
hypothesis  on  which  this  process  rests.  In  the  first  place  it  is  no 
hypothesis  that  we  may  conceive  the  waves  broken  up  on  arriving 
at  the  aperture  :  it  is  a  necessary  consequence  of  the  dynamical 
principle  of  the  superposition  of  small  motions,  and  if  this  principle 
be  inapplicable  to  light,  the  undulatory  theory  is  upset  from  its  very 
foundations.  The  mathematical  resolution  of  a  wave,  or  any  portion 
of  a  wave,  into  elementary  disturbances  must  not  be  confounded 
with  a  physical  breaking  up  of  the  wave,  with  which  it  has  no  more 
to  do  than  the  divisions  of  a  rod  of  variable  density  into  differential 
elements,  for  the  purpose  of  finding  its  centre  of  gravity,  has  to  do 
with  breaking  the  rod  in  pieces.  It  is  a  hypothesis  that  we  may 
find  the  disturbance  in  front  of  the  aperture  by  merely  taking  the 
aggregate  of  the  distubances  due  to  all  the  secondary  waves,  each 
secondary  wave  proceeding  as  if  the  screen  were  away  ;  in  other 
words,  that  the  effect  of  the  screen  is  merely  to  stop  a  certain  portion 
of  the  incident  light.  This  hypothesis,  exceedingly  probable,  a 
priori,  when  we  are  only  concerned  with  points  at  no  great  distance 
from  the  normal  to  the  primary  wave,  is  confirmed  by  experiment, 
which  shows  that  the  same  appearances  are  presented,  with  a  given 
aperture,  whatever  be  the  nature  of  the  screen  in  which  the  aperture 
is  pierced ;  whether,  for  example,  it  consist  of  paper  or  foil,  whether 
a  small  aperture  be  divided  by  a  hair  or  by  a  wire  of  equal  thickness. 
It  is  a  hypothesis,  again,  that  the  intensity  in  a  secondary  wave  is 
nearly  constant,  at  a  given  distance  from  the  centre,  in  different 
directions  very  near  the  normal  to  the  primary  wave  ;  but  it  seems 
to  me  almost  impossible  to  conceive  a  mechanical  theory  which 
would  not  lead  to  this  result.  It  is  evident  that  the  difference  of 
phase  of  the  various  secondary  waves  which  agitate  a  given  point 
must  be  determined  by  the  difference  of  their  radii,  and  if  it  should 


LIGHT  :    INTERFERENCE,  DIFFRACTION.  255 

afterwards  be  found  necessary  to  add  a  constant  to  all  the  phases 
the  results  will  not  be  at  all  affected.  Lastly,  good  reasons  may  be 
assigned  why  the  intensity  should  be  measured  by  the  square  of  the 
co- efficient  of  vibration.' 

225.  Huyghens'   construction,   if  rigorously   carried   out,  would 
indicate  the  existence  of  a  wave  running  back  towards  the  source 
as  well  as  a  wave  which  travels   forwards.      Analogy   points  to 
the  conclusion  that  the  part  of  the  construction  which  leads  to  a 
reverse  wave  must  be  ignored.     For  example,  the  investigation  of 
§  73  shows  that  no  wave  can  travel  backwards  from  a  disturbance 
which  runs  along  a  stretched  cord.     But  Stokes,  in  his  paper  on  the 
Dynamical  Theory  of  Diffraction,  of  the  introduction  to  which  the 
above  quotation  forms  part,  has  shown  from  purely  dynamical  prin- 
ciples, that  the  disturbance  in  a  secondary  wavelet  is  a  maximum 
in  the  direction  of  the  wave-normal,  and  that  it  diminishes  constantly 
as  the  direction  considered  is  inclined  more  and  more  to  the  normal, 
ultimately  becoming  zero  in  the  direction  opposite  to"  that  in  which 
the  primary  wave  travels.     He  then  shows  that  the  result  of  the 
superposition  of  all  the  secondary  effects  is  the  same  as  if  the  wave 
(assumed  to  be  practically  plane,  i.e.,  of  radius  which  is  large  in 
comparison  with  the  wave-length,  a  condition  always  satisfied  in 
experiment)  had  not  been  supposed  to  be  broken  up  into  a  series  of 
separate  centres   of   disturbance,   and  that  no  back-wave  is  pro- 
pagated. 

226.  Effect  of  a  Rectilinear  Wave. — We  have  already  stated  that 
Fresnel  showed  that  Huyghens'  principle,  according  to  which  the 
new  wave  front  is  found  to  be  the  envelope  of  the  secondary  wave- 
fronts,  should  be  explicitly  associated  with  the  principle  of  inter- 
ference if  it  is  to  give  a  complete  explanation  of  the  rectilinear  pro- 
pagation of  light.     The  envelope  is  the  locus  of  points  each  of  which 
is  simultaneously  reached  by  more  than  one  secondary  disturbance 
the  phases  of  which  are  identical.     It  is,  therefore,  the  locus  of 
points  at  which  the  light  has  great  intensity. 

The  necessity  for  the  introduction  of  the  principle  of  interference 
will  appear  very  evidently  from  the  following  investigation  of  the 
effect  of  a  rectilinear  wave  at  any  external  point. 

Let  AB  (Fig.  134)  represent  a  portion  of  a  linear  wave  which 
extends  to  infinity  in  both  directions,  and  let  P  be  the  point  at 
which  we  have  to  determine  the  effect  of  the  wave.  Draw  PM 
perpendicular  to  AB  and  take  points  m,  mr,  m",  etc.,  such  that 
Pra— PM  =  Pm'— Pm  =  Pm"-Pm'  =  etc.,  =  X/2,  where  X  is  the  wave- 
length of  the  light  emitted  from  the  various  points  of  AB. 

The  length  of  the  half-period  element  Mm  is   J 


256 


A    MANUAL    OF    PHYSICS. 


where  a,  is  the  length  of  PM.  When  X  is  so  small  in  comparison 
with  the  other  length  involved  that  it  may  be  neglected,  this 
becomes  ^/a\.  Similarly  Mm',  Mm",  etc.,  are  respectively  equal  to 


M 


FIG.  134. 


v/3»X,  etc.  Hence  the  lengths  of  the  successive  half-period 
elements,  from  M  outwards,  are  *Ja\,  ^/a\  (  */%  -  1),  */a\  (  *J3  —  ^/2), 
etc.,  and  the  limit  to  which  they  ultimately  approach  is  X/2. 

If  we  divide  each  element  into  the  same  number  of  infinitesi- 
mal portions,  the  light  sent  out  by  the  first  portion  of  the  first 
element  differs  in  phase  from  that  emitted  by  the  first  portion  of 
the  second  element  by  one  half  of  a  period.  Similarly,  the  light 
emitted  by  the  second  portion  of  the  first  element  differs  in  phase  by 
one  half  of  a  period  from  that  emitted  by  the  second  portion  of 
the  second  element,  and  so  on.  Now  the  effects  at  P  of  the  various 
parts  of  the  first  element  are  not  quite  compensated  by  the  effects 
of  the  corresponding  parts  of  the  second  element.  For  the  breadth 
of  the  parts  of  the  first  element  is  rather  greater  than  that  of  the 
parts  of  the  second  ;  and  the  inclination  of  each  part  to  the  line 
joining  it  to  P,  and  also  its  distance  from  P,  increase  as  the  part 
is  more  remote  from  M.  But  the  difference  between  the  effects  of 
the  corresponding  portions  of  the  rik  and  the  n+Vk  elements  is 
vanishingly  small  when  n  is  large. 

Now,  as  X  is  a  very  small  length  it  follows  that  a  very  large 
number  of  half  -period  elements  are  included  in  a  small  portion  of 
AB  in  the  near  neighbourhood  of  M,  and,  consequently,  only  a 
small  part  of  the  wave  near  M  produces  any  effect  at  P.  Hence,  a 
small  opaque  object  placed  on  the  line  PM  would  entirely  prevent 
the  wave  AB  from  producing  any  effect  at  P. 

Hence  the  propagation  of  light  is  practically  rectilinear  when  X  is 


LIGHT  :    INTERFERENCE,  DIFFRACTION.  257 

so  small  that  its  square  may  be  neglected  in  comparison  with  the 
other  quantities  involved. 

Let  e2,  <?2,  e*c-»  be  the  effects  at  P  of  the  first,  second,  etc.,  half- 
period  elements  of  MB.     The  total  effect  (taking  account  also  of  the 
portion  MA)  is 

2(ei  -  6^+63  -  e4+  ......  -  e»n  +  .....  )  . 

These  various  terms  are  in  descending  order  of  magnitude,  and 
therefore  it  appears  that  the  total  effect  is  smaller  than  the  effect  of 
the  first  half-period  portions  at  M. 

The  difference  between  any  two  successive  terms  is  small  in  com- 
parison with  the  magnitude  of  either,  and  so,  writing  the  above 
expression  in  the  form 


we  see  that  the  total  effect  at  P  is  approximately  equal  to  the  effect 
of  one  half  -period  element  at  M. 

227.  Effects  of  Plane  and  Spherical  Waves.  —  Let  the  plane  of 
the  paper  represent  a  plane  wave,  the  effect  of  which  at  a  point,  P, 
is  to  be  found,  and  let  M  (Fig.  135)  be  the  foot  of  the  perpendicular 
drawn  from  P  to  the  plane. 


FIG.  135. 

From  P  as  centre  describe  successive  spheres  of  radii  MP+X/2, 
MP+2X/2,  etc.  The  spheres  will  divide  the  plane  wave  into  con- 
centric zones,  called  half-period  zones,  or,  sometimes,  Huyghens' 
zones. 

Dividing  each  of  these  zones  into  the  same  number  of  infinitesi- 
mal annular  portions,  we  observe  that  the  effect  of  each  portion  of 
one  zone  is  nearly  annulled  by  the  effect  of  the  corresponding  por- 
tion of  the  succeeding  zone,  and  that  the  annulment  is  practically 
complete  at  a  short  distance  from  the  point  M — precisely  as  in  the 
similar  investigation  of  last  section.  Hence  the  effect  produced  at 

17 


258  A    MANUAL    OF    PHYSICS. 

P  is  that  due  to  a  few  half -period  zones  in  the  neighbourhood  of  the 
wave-normal  which  passes  through  P,  and  is  practically  equal  to 
half  the  effect  of  the  first  zone. 

Let  AMB  (Fig.  136)  represent  a  spherical  wave  diverging  from  O. 


FIG.  136. 

To  find  the  effect  at  P  we  must,  as  above,  divide  AMB  into  half-period 
zones  surrounding  M  the  point  in  which  OP  intersects  AB. 

Reasoning  similar  to  the  foregoing  shows  that  the  effect  of  the 
wave  at  P  is  equivalent  to  half  of  that  produced  by  the  first  zone. 

Now  it  is  easy  to  see  that  the  phase  of  the  vibration  due  to  the 
first  zone  differs  from  that  due  to  the  secondary  wave  at  M  by  one 
quarter  of  a  period.  For  if  OM  be  large  in  comparison  with  the 
wave-length — a  condition  which  is  satisfied  in  all  experimental 
observations — the  first  zone  is  practically  plane.  And,  further,  if  it 
be  broken  up  into  2n  infinitesimal  rings,  of  equal  area,  surrounding 
the  point  M  as  centre,  the  amplitudes  of  the  vibrations  produced 
at  P  by  each  of  these  annular  portions  will  be  practically  equal  to 
one  another.  Hence  (§  52)  the  phase  of  the  resultant  of  the  effects 
of  the  1"  and  the  2n'A  annuli  is  halfway  between  those  of  its  com- 
ponents. This  is  true  also  of  the  phase  of  the  resultant  of  the 
effects  of  the  2"rf  and  the  (2n— 1)'*  annuli,  and  so  on.  But  the  phase 
of  the  vibration  at  P  due  to  each  annulus  varies  uniformly  from  the 
1"  to  the  2wM  annulus.  Therefore  the  phase  of  the  resultant  vibra- 
tion at  P  due  to  the  complete  zone  is  one-quarter  of  a  period  behind 
that  due  to  the  vibration  at  M.  The  same  statement  must  be  true 
of  the  vibration  produced  by  the  whole  wave  if  it  agrees  in  phase 
with  that  produced  by  the  first  zone. 

228.  Diffraction  at  a  Straight  Edge. — We  are  now  in  a  position 
to  determine  the  effects  produced  by  any  given  portions  of  a  wave 
which  diverges  from  a  luminous  point — the  remaining  portions 
being  intercepted  by  opaque  obstacles.  This  involves  the  carrying 
of  our  investigations  beyond  the  stage  in  which  the  wave-length 
may  be  assumed  to  be  small  in  comparison  with  all  other  quantities 


LIGHT  :    INTERFERENCE,  DIFFRACTION. 


259 


involved.  We  shall  find  that,  under  this  new  condition,  light  is  no 
longer  propagated  in  straight  lines,  but  is  bent,  or  diffracted,  into 
the  geometrical  shadows  precisely  as  sound  is. 

AMB  (Fig.  137)  represents  a  spherical  wave,  which,  diverging  from 
the  point  0,  is  partially  intercepted  by  an  opaque  object  MN.  We 
have  to  determine  (1)  the  effect  at  any  point  P  outside  the  geo- 


B 


FIG.  137. 

metrical  boundary,  OMC,  of  the  shadow  ;  (2)  the  effect  at  any  point 
inside  Q  the  geometrical  shadow. 

Join  OP  and  MP,  and  let  OP  meet  AB  in  ra. 

When  mM  contains  a  considerable  number  of  half-period  elements, 
the  wave  produces  practically  its  full  effect  at  P.  Let  elt  e2,  etc.,  be 
the  effects  of  the  first,  second,  etc.,  half  -period  elements  in  the 
neighbourhood  of  m  ;  and  let  E  be  the  effect  of  the  semi-wave 
raB.  The  effect  at  P  is  E,  E+el5  E  •}-e1  -  e2,  etc.,  according  as  the 
number  of  elements  included  in  mM  is  0,  1,  2,  etc.  Hence  the 
effect  at  P  is  a  maximum,  or  a  minimum,  according  as  mM  contains 
an  odd,  or  an  even,  number  of  half  -period  elements  ;  that  is,  accord- 
ing as,  in  the  formula 


n  is  odd  or  even.     When  n  and  A  are  given,  the  locus  of  P  is  a 
hyperbola  the  foci  of  which  are  0  and  M.     Now,  if  we  denote  PC  by 

and  MP  = 


x,  OM  by  a,  and  MC  by  6,  we  get 

&2-f-#2'  These  expressions  give  approximately 

and  MP  =  6-f  #2/26.     Hence  the  above  formula  becomes 


At  a  point  Q,  within  the  geometrical  shadow,  the  most  effective 

17—2 


260 


A    MANUAL    OF    PHYSICS. 


portions  of  the  wave  are  intercepted  by.  the  obstacle.  The  effect  at 
Q  is  practically  \e*,  \e^  etc.,  according  as  MN  intercepts  1,  2,  etc., 
of  the  most  powerful  elements.  Hence  the  illumination  inside  the 
geometrical  shadow  dies  away  as  the  distance  of  the  illuminated 
point  from  the  geometrical  boundary  increases. 

Diffraction  fringes,  resembling  those  just  described,  appear  out- 
side the  geometrical  shadow  on  both  sides  of  a  narrow  obstacle, 
such  as  a  thin  wire  or  a  hair.  But,  in  addition,  a  series  of  finer  bands, 
of  constant  breadth,  make  their  appearance  inside  the  geometrical 
shadow  if  the  obstacle  is  sufficiently  narrow.  These  are  caused  by 
interference  of  the  light  diffracted  at  both  sides  of  the  obstacle,  for, 
as  we  have  seen,  the  effect  of  each  unintercepted  portion  of  the 
wave  is  practically  the  same  as  that  of  a  luminous  line  placed  close 
to  the  straight  edge  of  the  obstacle. 

229.  Diffraction  at  a  Narrow  Slit. — Let  MN  (Fig.  138)  represent 
a  narrow  opening  in  an  opaque  obstacle,  and  let  a  wave  AB 
diverge  from  a  point  O,  which  is  situated  on  the  line  drawn  from  the 
middle  point  of  MN  at  right  angles  to  the  plane  of  the  obstacle. 


FIG.  138. 

Reasoning  similar  to  that  of  last  section  shows  that  the  illumina- 
tion at  a  point  P  will  be  a  maximum,  or  a  minimum,  according  as 
MN  contains  an  odd,  or  an  even,  number  of  half-period  elements. 

Let  M'N'  be  the  geometrical  projection  of  MN.  If  the  screen, 
PM'N',  be  so  far  from  MN  that  NM'-MM'  (or  MN'-NN')  is  less 
than  a  semi-wave-length,  a  fringe  of  alternately  bright  and  dark 
bands  will  appear  on  each  side  of  the  geometrical  projection  of 
MN.  But  if  the  distance  between  the  obstacle  and  the  screen  be  so 
small  that  NM'  -  MM'  is  greater  than  a  semi-wave-length,  bands 
will  appear  between  M'  and  N'. 

230.  Diffraction  at  a  Circular  Aperture.     Zone  Plates. — Draw 


LIGHT  I    INTERFERENCE,  DIFFRACTION.  261 

the  line  OP  (Fig.  139)  through  the  centre  of  the  aperture  and  per- 
pendicular to  its  plane.  We  shall  determine  the  general  effect  at 
P  of  light  diverging  from  0. 


FIG.  139. 

The  illumination  at  P  is  a  maximum,  or  a  minimum,  according 
as  MN  contains  an  odd,  or  an  even,  number  of  half -period  zones. 

Let  a  be  the  centre  of  the  aperture,  and  let  b  be  the  outer  edge  of 
the  nth  zone.  Denote  the  lengths  of  Oa,  #P,  and  ab  by  u,  v,  and  x 
respectively.  We  get  approximately 


and 

therefore 

Hence 

But  this  length  is  equal  to  w\/2,  and  so 


Thus  the  consecutive  values  of  x  are  proportional  to  the  square 
roots  of  the  natural  numbers. 

As  P  approaches  MN,  the  number  of  half -period  zones  in  the 
aperture  increase,  and  so  the  illumination  at  P  passes  through  a 
succession  of  maxima  and  minima.  The  various  points  at  which 
the  maxima  and  minima  occur  are  given  by  the  expression 

ur2 

v  =  —     — 9 
un\— r2 

where  r  is  the  radius  of  the  aperture. 


262  A   MANUAL   OF   PHYSICS. 

From  the  way  in  which  X  is  involved,  it  is  evident  that  the 
position  of  P  corresponding  to  maximum  illumination  approaches 
nearer  to  the  aperture  when  the  wave-length  increases  ;  so  that  an 
eye,  which  advances  to  the  aperture  along  Pa,  will  perceive  a  rapid 
periodic  variation  in  the  colour  of  the  light  which  reaches  it. 

If,  as  formerly  (§  228),  we  denote  by  e1}  e2,  etc.,  the  effects  of  the 
light  which  passes  through  the  successive  half-period-zones,  the 
total  effect  is 


The  effect  will  therefore  be  much  greater  than  it  otherwise  could 
be  if  the  even  zones  be  made  opaque.  Such  an  arrangement  con- 
stitutes a  zone  plate.  If  n  be  the  number  of  zones  (alternately 
open  and  opaque)  in  a  zone  plate  of  radius  r,  the  formula 

1     l_nX 

u    v~  r'2 

shows  (§  193)  that  the  plate  acts  as  a  condensing  lens,  the  principal 
focal  length  of  which  is  r2/nX.  But,  in  the  case  of  the  zone  plate, 
all  rays  do  not  take  the  same  time  to  pass  between  the  conjugate 
foci  ;  and,  further,  the  focus  for  red  rays  is  nearer  to  the  plate  than 
the  focus  for  blue  rays  is.  In  these  points  it  differs  from  a  lens. 

231.  Diffraction  at  an  Opaque  Disc.  —  A  point  at  the  centre  of 
the  geometrical  shadow  is  almost  as  brightly  illuminated  as  if  the 
disc  were  removed.     If  this  disc  removes  n  -  1  half  -period  zones, 
the  effect  of  the  remaining  zones  is  practically  equal  to  one  half  of 
that  of  the  wth  zone.     But,  so  long  as  n  is  not  large,  the  effect  of  the 
wth  disc  is  not  greatly  different  from  that  of  the  first.     This  theo- 
retical result  was  first  pointed  out  by  Poisson,  and  was  verified 
experimentally  by  Arago. 

232.  Cor  once.     Young'  s  Eriometer.  —  If  a  number  of  very  small 
and  nearly  equal  particles  be  closely  distributed  in  the  space  inter- 
vening between  a  luminous  object  and  the  eye,  the  object  will  appear 
to  be  surrounded  by  luminous  rings.     These  are  due  to  diffraction  of 
the  light  which  passes  the  edges  of  the  particles.    The  coronce,  which 
are  sometimes  seen  surrounding  the  sun  or  the  moon,  are  caused  by 
the  presence  of  small  globules  of  water  in  the  atmosphere.     They 
are  coloured  blue  inside,  red  outside,  and  increase  in  size  when  the 
diameters  of  the  globules  diminish.      [If,  therefore,  the  corona3  are 
observed  to  contract,  the  moisture  in  the  atmosphere  is  condensing, 
and  rain  may  be  expected  to  follow  ;  conversely,  if  the  rings  dilate, 
dry  weather  will  in  general  ensue.] 

Young's  Eriometer  was  devised  for  the  purpose  of  measuring  the 


LIGHT  I    INTERFERENCE,  DIFFRACTION.  263 

diameters  of  small  objects.  It  consists  of  a  metal  plate,  in  which  a 
small  hole  is  drilled.  The  plate  is  also  perforated  by  a  circle  of  still 
smaller  holes  which  surround  the  large  hole  as  a  centre.  A  flame  is 
placed  behind  this  plate,  and  the  light  which  passes  through  the 
holes  is  examined  through  glass  plates  which  contain  between  them 
the  (equal)  particles  the  size  of  which  is  to  be  determined.  The  large 
opening  in  the  metal  plate  is  surrounded  by  coloured  rings,  and  the 
distance  between  the  metal  plate  and  the  glass  plates  is  altered  until 
any  one  particular  ring  coincides  with  the  circle  of  small  holes. 
This  distance  varies  inversely  as  the  radius  of  the  ring,  which  itself, 
as  we  have  just  seen,  varies  inversely  as  the  diameter  of  the 
particles.  One  experiment,  in  which  the  diameter  of  the  particles 
is  known,  and  the  distance  between  the  plates  is  measured,  is 
sufficient  to  enable  us  to  calculate  the  unknown  diameters  of  other 
sets  of  particles. 

233.  Diffraction  Gratings. — A  diffraction  grating  may  consist  of 
a  glass  plate,  upon  which  a  great  number  of  extremely  fine  equi- 
distant parallel  lines  are  ruled  by  means  of  a  diamond  point.  The 
grooves  are  practically  opaque,  for  light  incident  on  them  is  reflected 
back  in  all  directions.  On  the  other  hand,  the  glass  between  the 
grooves  is  transparent,  and  the  light  which  passes  through  is 
diffracted  in  all  directions. 

Let  AB  (Fig.  140)  represent  a  (highly-magnified)  portion  of  the 
grating,  the  dark  parts  indicating  the  grooves,  and  the  light  parts 
indicating  the  intervening  spaces ;  and  let  aP  and  6P  represent  the 


FIG.  140. 

paths  of  rays  which  reach  P  from  similarly-situated  parts  of  the 
openings  a  and  6.  From  a  drop  a  perpendicular  am  upon  6P. 
The  distance  ab  being  very  small  in  comparison  with  the  distance 
of  P  from  the  grating,  bm  is  practically  equal  to  6P  -  aP. 

Now  suppose  that  parallel  rays  from  a  narrow  slit  parallel  to  the 
grooves  fall  perpendicularly  upon  the  grating.  An  eye  placed  at  P 
will  see  the  slit  through  the  grating  in  the  direction  PQ.  The 


264  A    MANUAL    OF    PHYSICS.  , 

angle  bam  is  equal  to  the  angle  aPQ(  =  0,  say)  ;  and  so  bm  =  ab  sin0. 
Therefore  a  maximum  or  minimum  effect  will  be  produced  at  P 
according  as  n  is  even  or  odd  in  the  expression 


The  length  ab  is  known,  since  it  is  the  reciprocal  of  the  number 
of  grooves  ruled  in  unit  breadth  of  the  grating. 

If  monochromatic  light  be  used,  a  series  of  coloured  images  of 
the  slit  will  be  seen  at  different  angular  distances  from  the  line 
PQ.  If  white  light  be  used,  a  series  of  spectra  will  be  observed,  in 
each  of  which  the  violet  light  is  less  bent  from  its  original  direction 
than  the  red  light  is.  The  spectra  are  said  to  be  of  the  first,  second, 
etc.,  order,  according  as  n  has  the  values  1,  2,  etc.  All  the  spectra 
beyond  the  second  partially  overlap  each  other. 

Very  accurate  measurements  of  wave-length  may  be  made  by 
means  of  the  grating  ;  and  the  spectra  obtained  from  all  gratings 
are  identical,  except  as  regards  scale  ;  that  is,  there  is  no  trace  of 
irrationality  in  the  dispersion  (§  197).  And,  further,  if  9  is  nearly 
zero,  the  dispersion  between  any  two  rays  is  practically  proportional 
to  the  difference  of  their  wave-lengths.  This  condition  may  be 
attained  by  inclining  the  grating  to  the  direction  of  the  incident 
light  at  a  suitable  angle.  The  spectrum  thus  produced  is  called 
a  normal  spectrum. 

Diffraction  spectra  may  be  obtained  by  reflection  from  a  ruled 
metallic  surface.  Eowland's  concave  gratings  are  ruled  on  the 
polished  surface  of  a  portion  of  a  cylinder  of  speculum  metal. 


CHAPTEE  XIX. 

DOUBLE    REFRACTION.       POLARISATION. 

234.  Double  Eefraction. — In  our  consideration  of  the  refraction  of 
light,  we  have  hitherto  dealt  only  with  those  cases  in  which  a  single 
refracted  ray  occurs. 

Bartholinus,  in  1669,  described  the  phenomenon  of  double  refrac- 
tion as  observed  by  him  in  Iceland  spar. 

A  single  ray  of  light  incident  upon  the  surface  of  Iceland  spar  in 
general  gives  rise  to  two  refracted  rays.  One  of  these  obeys  the 
ordinary  law  of  refraction,  but  the  other  follows  a  totally  different 
law.  The  former  is  called  the  ordinary,  and  the  latter  the  extra- 
ordinary, ray. 

All  crystalline  minerals,  except  those  belonging  to  the  cubic 
system,  possess  the  property  of  double  refraction. 

The  fundamental  form  in  which  Iceland  spar  crystallizes  is  the 
rhombohedron.  The  angles  of  the  faces  are  either  acute  or  obtuse. 
The  obtuse  angles  are  all  equal  to  each  other,  and  the  acute  angles 


are  all  equal  also.  Two  of  the  solid  angles  (A  and  B,  Fig.  141)  of  the 
rhombohedron  are  bounded  by  three  obtuse  angles.  All  other  angles, 
such  as  C,  are  bounded  by  one  obtuse  and  two  acute  angles.  The 
axis  of  the  crystal  is  a  line  which  is  equally  inclined  to  the  three 
edges  meeting  at  an  obtuse-angled  corner.  If  we  make  all  the 
edges  of  the  block  equal  in  length,  the  crystalline  axis  will  be  AB, 


266  A   MANUAL    OF   PHYSICS.  , 

the  diagonal  joining  the  two  obtuse-angled  corners.  A  plane  ACB, 
which  passes  through  the  crystalline  axis  and  the  shorter  diagonal 
AC  of.  a  rhombic  face,  is  called  a,  principal  section  of  the  crystal. 

In  Iceland  spar,  and  in  many  other  crystalline  substances,  all  the 
optical  properties  are  symmetrical  about  the  axis  of  form.  Any 
direction  in  such  a  substance,  which  is  parallel  to  the  axis  of  form, 
is,  therefore,  called  the  optic  axis ;  and  all  such  substances  are 
called  uniaxal  crystals. 

If  the  spar  be  cut  by  a  plane  in  any  direction,  and  a  ray  of  light 
falls  upon  the  surface  so  formed,  both  an  ordinary  and  an  extra- 
ordinary ray  will  in  general  be  produced ;  and,  in  most  cases,  the 
latter  will  not  lie  in  the  plane  of  incidence.  But  if  the  plane  be 
perpendicular  to  the  optic  axis,  both  rays  coincide  if  the  incidence 
is  normal.  This  also  occurs  if  the  optic  axis  lies  in  the  refracting 
surface,  and  the  incidence  is  normal ;  and,  further,  in  this  case  the 
extraordinary  ray  obeys  the  ordinary  law  so  long  as  the  plane  of 
incidence  is  perpendicular  to  the  optic  axis. 

These  various  phenomena  were  investigated  very  fully  by 
Huyghens,  and  he  was  led  to  adopt  a  construction  for  the  wave-front 
in  the  interior  of  the  crystal  which  he  himself  proved  experiment- 
ally to  accord  very  accurately  with  the  observed  facts.  More  severe 
tests  of  his  construction  were  made  by  Wollaston  in  1802 ;  and 
recently  Stokes,  Mascart,  and  Glazebrook  have  verified  its  accuracy 
to  the  full  extent  attainable  by  modern  methods  of  measurement. 

235.  Huyghens^  Construction. — Huyghens  had  previously  ex- 
plained the  propagation  of  light  in  homogeneous  isotropic  media  by 


FIG.  142. 

the  assumption  that  the  wave-surface  was  spherical  (§§  186,  200). 
To  explain  double  refraction  in  uniaxal  crystals,  he  assumed  that 
the  wave-surface  consists  of  an  ellipsoid  of  revolution  the  axis  of 
symmetry  of  which  is  coincident  with  the  optic  axis,  and  a  sphere 
which  touches  the  ellipsoid  at  the  extremity  of  its  axis  of  symmetry. 


LIGHT  :    DOUBLE    REFRACTION,    POLARISATION.  267 

The  spherical  portion  of  the  surface  corresponds  to  uniform  speed  of 
propagation  in  all  directions ;  and  the  incident  ray  being  given? 
we  can  determine  from  it,  by  the  method  of  §  200,  the  direction 
of  the  ordinarily  refracted  ray.  The  ellipsoid  indicates  unequal 
speed  of  propagation  in  different  directions,  and  from  it  we  can 
determine  the  direction  of  the  extraordinarily  refracted  ray  by  a 
similar  process. 

Let  0  (Fig.  142)  be  the  point  at  which  an  incident  ray  AO  meets 
the  surface  OQ,  and  let  BPQ  be  another  ray  parallel  to  AO,  so  that 
OP,  which  is  perpendicular  to  both,  may  represent  a  portion  of  a 
plane  wave-front.  In  the  time  in  which  light  moves  from  P  to  Q, 
the  ordinary  ray  will  have  passed  over  a  distance  OB,  such  that  PQ  = 
juOB,  where  p  is  the  ordinary  index  of  refraction.  A  plane  through 
Q,  perpendicular  to  the  plane  of  incidence,  will  touch  a  sphere  drawn 
from  0  with  radius  OB  in  a  point  B,  and  OB  is  the  direction  of 
the  ordinary  ray.  The  plane  BQ  is  the  ordinarily  refracted  wave- 
front. 

If  00  is  the  optic  axis,  the  radii  of  an  ellipsoid  OS,  which  has 
00  as  its  semi-diameter  of  revolution,  will  represent  the  speeds  of 
propagation  of  the  extraordinary  ray  in  different  directions.  A 
plane  passing  through  Q,  and  perpendicular  to  the  plane  of  inci- 
dence, will  touch  OS  in  a  point  S  such  that  OS  is  the  direction  of 
the  extraordinary  ray ;  and  the  ratio  PQ/OS  is  equal  to  /*',  the  index 
of  refraction  for  all  extraordinary  rays  which  pass  through  the 
crystal  in  the  direction  OS. 

In  Iceland  spar,  OC  is  the  shortest  radius  of  the  ellipsoid ;  in 
quartz  it  is  the  largest  radius.  All  crystals  which  resemble  Iceland 
spar  in  this  respect  are  called  negative  crystals ;  those  which  resemble 
quartz  are  called  positive  crystals.  In  the  former,  the  extraordinary 
index  is  less  than  the  ordinary;  in  the  latter,  the  reverse  is  the 
case. 

If,  in  this  figure,  the  point  C  lies  out  of  the  plane  of  the  paper,  the 
point  S  will  in  general  lie  outside  it  also  ;  that  is,  the  extraordinary 
ray  will  not  be  in  the  plane  of  incidence.  This  will  be  so  even  if 
the  incidence  is  perpendicular ;  for  the  new  wave-front  will  be  a 
plane  parallel  to  OQ,  and  this  will  in  general  touch  CS  in  a  point 
which  does  not  lie  in  the  plane  of  the  paper. 

236.  Special  Sections  of  the  Surface. — (1)  Let  the  refracting 
surface  be  perpendicular  to  the  optic  axis  (Fig.  143).  At  normal 
incidence  there  is  no  separation  of  the  two  rays  ;  but,  as  the  angle 
of  incidence  increases,  the  extraordinary  ray  separates  out  farther 
from  the  normal  than  the  ordinary  one  does.  If  the  plane  of 
incidence  be  rotated  around  OC,  the  two  rays  each  maintain  a 


268 


A   MANUAL    OF    PHYSICS. 


fixed  inclination  so  long  as  the  angle  of  incidence  remains  con- 
stant. 


FIG.  143. 

(2)  Let  the  refracting  surface  and  the  plane  of  incidence  intersect 
in  the  optic  axis  (Fig.  144).  At  normal  incidence  there  will  be  no 
separation  of  the  two  rays  as  regards  direction,  though  the  extra- 
ordinary ray  will  travel  with  greater  speed  than  will  the  ordinary 


FIG.  144. 

ray.  And,  when  the  angle  of  incidence  increases,  the  former  does 
not  separate  out  so  far  from  the  normal  as  the  latter  does  ;  for, 
from  the  properties  of  the  ellipse  and  circle  with  a  common  diameter, 
R  and  S  lie  on  a  line  which  is  perpendicular  to  0  Q. 


FIG.  145. 

(3)  Let  the  refracting  surface  contain  the  optic  axis,  while  the  plane 
of  incidence  is  perpendicular  to  it  (Fig.  145).     The  section  of  the  ellip- 


LIGHT  :    DOUBLE    REFRACTION,    POLARISATION.  269 

sold  becomes  a  circle,  and  so  the  extraordinary  ray  obeys  the  ordinary 
law,  though  its  index  of  refraction  is  less  than  that  of  the  ordinary 
ray.  At  normal  incidence,  this  case  becomes  identical  with  the 
last. 

237.  Polarisation. — Huyghens  observed  that  the  intensities  of  the 
two  beams  produced  by  refraction  in  a  block  of  Iceland  spar  are 
equal.  And  he  further  noticed  that  each  of  these  beams  was  in 
general  subdivided  into  two  others,  of  unequal  intensity,  on  trans- 
mission through  a  second  block. 

When  the  principal  sections  of  the  two  blocks  are  parallel,  no 
more  than  two  beams  are  produced :  the  ordinary  ray  in  the  first 
block  passes  through  the  second,  without  change  of  direction,  as  an 
ordinary  ray  ;  and  the  extraordinary  ray  passes  through  also  with- 
out any  change.  And,  when  the  principal  sections  of  the  blocks  are 
at  right  angles  to  each  other,  two  rays  only  are  transmitted ;  but  the 
ordinary  ray  in  the  first  block  passes  through  the  second  as  an  extra- 
ordinary ray,  while  the  extraordinary  ray  in  the  first  becomes  an 
ordinary  ray  in  the  second.  In  all  other  relative  positions  of  the 
two  principal  sections,  each  ray  in  the  first  is  subdivided  into  two 
in  the  second.  As  the  second  block  is  turned  round  from  the 
position  in  which  its  principal  section  was  parallel  to  that  of  the 
first,  the  two  original  beams  gradually  diminish  in  intensity  as  the 
intensities  of  the  newly-produced  beams  increase.  When  the  prin- 
cipal sections  are  inclined  at  an  angle  of  45°  to  each  other,  all  the 
four  rays  are  equally  intense.  The  changes  then  proceed  in  the 
same  direction  until  the  inclination  of  the  principal  sections  is  90°, 
when  the  original  beams  vanish ;  after  this,  if  the  inclination  be 
still  further  increased,  the  changes  proceed  in  the  reverse  order 
until,  at  180°,  the  beams  again  pass  unchanged  through  the  second 
block. 

Huyghens  remarked  that  the  rays  which  had  passed  through  the 
first  block  seemed  to  have  acquired  some  form  or  disposition 
which  led  to  the  production  of  these  phenomena.  Newton  spoke  of 
them  as  possessing  sides.  But  it  was  not  until  more  than  a  cen- 
tury afterwards  that  a  complete  explanation  was  found  as  the  result 
of  an  accidental  discovery. 

Malus,  happening  to  examine  through  a  doubly  refracting  prism 
the  light  reflected  from  the  windows  of  the  Luxembourg  Palace, 
observed  that  each  ray  alternately  disappeared  as  he  rotated  the 
prism  through  successive  angles  of  90°.  He  said  that  the  light  was 
polarised  ;  for,  favouring  the  corpuscular  theory,  he  concluded  that 
the  corpuscles  possessed  poles,  which  gave  rise  to  the  observed 
effects.  (The  plane  of  reflection  of  the  polarised  light  is  called  the 


270  A    MANUAL    OF    PHYSICS. 

plane  of  polarisation.}  Extending  his  investigation,  he  found  that 
the  light  which  is  reflected  from  the  surface  of  any  transparent 
medium,  at  definite  angles  (called  the  angles  of  polarisation)  which 
depend  upon  the  nature  of  the  medium,  exactly  resembles  one  of 
the  beams  which  have  passed  through  a  doubly  reflecting  sub- 
stance. 

The  reflected  light  has  the  same  properties  as  the  ordinary  ray  in 
Iceland  spar  has  if  the  plane  in  which  it  is  reflected  is  parallel  to 
the  optic  axis  of  the  spar ;  it  manifests  the  properties  of  the  ex- 
traordinary ray  if  its  plane  of  reflection  is  perpendicular  to  the 
axis. 

The  supporters  of  the  undulatory  theory  at  first  regarded  the 
vibrations  as  taking  place  in  the  direction  in  which  the  waves 
travelled,  but  the  phenomena  of  polarisation  cannot  be  explained  on 
this  assumption.  In  particular,  the  conditions  which  are  essential 
to  the  production  of  interference  of  polarised  light  (§  247)  necessitate 
the  assumption  that  the  vibrations  take  place  perpendicularly  to  the 
direction  of  the  ray. 

Hooke,  in  1672,  had  suggested  that  the  vibrations  occur  in  direc- 
tions which  are  perpendicular  to  the  ray ;  but  the  idea  was  never 
developed  until  its  truth  was  inferred  by  Young  and  Fresnel,  inde- 
pendently, not  long  after  Malus  had  discovered  that  light  was 
capable  of  undergoing  polarisation  by  reflection. 

A  ray  of  light  in  which  the  vibrations  of  the  ether  all  take 
place  in  one  common  direction,  evidently  possesses  l  form'  or  '  dis- 
position^ or  '  sides.''  Think,  for  example,  of  a  stretched  cord  placed 
between  two  smooth  parallel  planes  which  just  touch  it.  Waves 
in  which  the  vibrations  are  parallel  to  these  planes  can  pass  along 
the  cord ;  perpendicular  vibrations  are  incapable  of  existing.  The 
waves  possess  '  sides  '  which  are  in,  and  perpendicular  to,  the  planes 
which  confine  the  cord. 

238.  Laws  of  Polarisation  by  Reflection  and  Refraction. — (1) 
Brewster's  Law.  Brewster  made  an  elaborate  series  of  investiga- 
tions on  the  angles  of  polarisation  of  various  substances,  with  the 
object  of  connecting  the  phenomenon  with  other  optical  properties 
of  the  substances.  He  found  that  the  index  of  refraction  is  equal 
to  the  tangent  of  the  angle  of  polarisation. 

From  this  law  we  can  at  once  deduce  the  relation  cos  i=sin  r, 
when  i  and  r  are  the  angles  of  incidence  and  refraction.  Hence, 
the  refracted  ray  is  perpendicular  to  the  reflected  ray. 

Since  the  refractive  index  varies  with  the  wave-length,  rays  of 
different  colours  are  polarised  at  different  angles. 

Jamin   has   found   that   the  polarisation   is  not   quite   complete 


LIGHT  :    DOUBLE    REFRACTION,  POLARISATION.  271 

except  in  some  substances  the  refractive  index  of  which  is  about 
1-46. 

(2)  Arago's  Law.     The  light  which  is  refracted  into  a  transparent 
medium  is  polarised  to  a  greater  or  less  extent.     Arago  found  that 
the  quantity  of  polarised  light  in  the  refracted  beam  is  equal  to 
the  quantity  in  the  reflected  beam,  and  the  planes  of  polarisation 
of  the  two  are  at  right  angles  to  each  other. 

Brewster's  Law  is  applicable  to  reflection  in  the  interior  of  a  dense 
substance.  Hence  part  of  the  unpolarised  light  in  the  refracted 
beam  will  undergo  further  polarisation  when  it  is  reflected  at  the 
second  surface  of  the  substance.  If  a  sufficient  number  of  parallel 
reflecting  surfaces,  such  as  those  of  a  number  of  thin  plates  of  glass 
placed  one  behind  the  other,  be  provided,  the  incident  light  may  be 
divided  into  a  reflected  and  a,  refracted  beam,  each  of  which  is 
totally  polarised  in  a  plane  at  right  angles  to  the  plane  of 
polarisation  of  the  other.  This  arrangement  constitutes  a  '  pile  of 
plates.' 

(3)  Mains' s  Law.    Light,  which  is  incident  at  the  polarising  angle 
on  a  plane  reflecting  surface,  is  totally  unaffected,  as  regards  inten- 
sity, by  a  second  reflection  from  a  parallel  plate  of  the  same  sub- 
stance.    But,  if  the  plane  of  incidence  upon  the  second  plate  be 
perpendicular  to  the  plane  of  reflection  from  the  first,  the  reflected 
beam  will  be  totally  extinguished.    This  subject  was  fully  investigated 
by  Malus,  who  found  that  the  intensity  of  the  twice-reflected  beam 
is  proportional  to  the  square  of  the  cosine  of  the  angle  of  inclina- 
tion of  the  two  planes  of  reflection. 

239,  Direction  of  Vibration  in  Polarised  Light. — The  reflected 
ray,  which  (according  to  definition)  is  polarised  in  the  plane  of 
reflection,  has  symmetry  with  regard  to  that  plane,  since  its  inten- 
sity is  totally  unaltered  by  any  number  of  reflections  in  that  plane. 
It  has  also  symmetry  with  regard  to  the  normal  to  the  plane  of 
reflection,  since  it  vanishes  on  reflection  in  any  plane  which  passes 
through  this  normal. 

We  may  therefore  assume  either  that  the  direction  of  the  vibra- 
tions in  polarised  light  is  perpendicular  to  the  plane  of  reflection,  or 
that  it  lies  in  the  plane  of  reflection.  Fresnel,  in  his  theoretical 
investigations,  made  the  former  assumption;  Maccullagh  and 
Neumann  adopted  the  latter. 

The  truth  of  the  former  is  indicated  by  a  number  of  considera- 
tions. 

The  vibrations  of  the  ordinary  ray  in  Iceland  spar  will  be  perpen- 
dicular to  the  optic  axis  provided  that  the  vibrations  of  a  ray 
polarised  by  reflection  are  perpendicular  to  the  plane  of  polarisation  ; 


272  A    MANUAL    OF    PHYSICS. 

and  thus  the  uniform  speed  of  that  ray,  in  all  directions,  is  readily 
accounted  for.  But,  on  the  alternative  assumption,  the  pro- 
perties of  the  ordinary  ray  would  be  exceedingly  difficult  of  explana- 
tion. 

The  ordinary  and  extraordinary  rays  produced  by  transmission 
through  certain  crystals,  such  as  tourmaline,  are  coloured.  "When 
the  two  rays  pursue  nearly  the  same  paths,  identical  colours  are 
exhibited  in  each ;  and  this  occurs  when  the  two  rays  traverse  the 
substance  nearly  in  the  direction  of  the  optic  axis,  so  that  their 
vibrations  are  nearly  perpendicular  to  it.  As  the  rays  separate  out 
from  the  axis,  the  colour  of  the  ordinary  ray  remains  constant,  while 
that  of  the  extraordinary  changes  greatly.  Haidinger  remarked  that 
this  favours  the  assumption  that  the  vibrations  of  the  ordinary  ray 
are  normal  to  the  optic  axis,  and  therefore  take  place  along  the 
normal  to  the  plane  of  polarisation. 

If  a  horizontal  beam  of  polarised  light,  the  vibrations  of  which 
are  in  lines  inclined  at  an  angle  a  to  the  vertical,  falls  perpendicularly 
on  a  diffraction  grating,  the  lines  of  which  are  vertical,  the  direction 
of  vibration  in  the  diffracted  beam  will  make  with  the  vertical  an  angle 
j3,  which  differs  from  a.  Let  a  be  the  amplitude  of  the  incident 
vibration.  The  resolved  part  of  it  parallel  to  the  lines  of  the  grating 
is  a  cos  a  ;  and  the  part  at  right  angles  to  this  is  a  sin  a.  If  the 
diffracted  beam  makes  an  angle  0  with  the  normal  to  the  grating, 
the  part  of  a  sin  a,  which  is  perpendicular  to  the  diffracted  beam,  is 
a  sin  a  cos  0,  and  it  is  this  part  alone  which  is  effective  in  the  propa- 
gation of  light.  Hence  the  tangent  of  the  angle  which  the  new 
direction  of  vibration  makes  with  the  lines  of  the  grating  is 
tan  J3  =  a  sin  a  cos  <j)ja  cos  a  =  tan  a  cos  0.  The  angle  /3  is  therefore 
less  than  a.  Consequently,  if  the  plane  of  polarisation  is  perpendi- 
cular to  the  direction  of  vibration,  the  plane  o^f  polarisation  of  the 
diffracted  beam  will  be  more  nearly  perpendicular  to  the  lines  of  the 
grating  than  that  of  the  incident  beam  ;  and  the  reverse  will  happen 
if  the  direction  of  vibration  lies  in  the  plane  of  polarisation. 

This  result  was  deduced  from  theory  by  Stokes.  He  also  tested  it 
experimentally,  and  found  that  the  result  seemed  to  support 
Fresnel's  assumption. 

Another  test,  also  due  to  Stokes,  is  based  upon  the  nature  of  the 
polarisation  of  light  which  has  undergone  reflection  from  very  small 
material  particles. 

Stokes  remarks  that  no  conclusion  can  be  drawn  so  long  as  the 
particles  are  large  compared  with  the  wave-length  of  light,  for  then 
reflection  occurs  as  it  would  from  the  surface  of  a  large  solid ;  but, 
when  the  particles  are  small  compared  with  the  wave-length,  it 


LIGHT  :    DOUBLE    REFRACTION,  POLARISATION.  273 

seems  plain  that  the  vibrations  in  the  incident  and  the  reflected  rays 
cannot  be  at  right  angles  to  each  other. 

The  small  particles  with  which  he  experimented  were  obtained  by 
highly  diluting  some  tincture  of  turmeric  with  alcohol  and  adding 
water.  A  horizontal  beam  of  sunlight  fell  upon  the  particles,  and 
the  light  was  found  to  be  polarised  in  the  plane  of  reflection.  The 
smaller  the  particles  were,  the  greater  was  the  tendency  to  complete 
polarisation  in  the  plane  of  reflection. 

Since  the  '  sides '  of  the  reflected  ray  are  symmetrical  with 
respect  to  the  plane  of  polarisation,  its  vibrations  must  either  be 
parallel  to  the  incident  ray  or  perpendicular  to  the  plane  of  reflec- 
tion, i.e.,  of  polarisation.  We  must  therefore  choose  the  latter 
alternative,  since  we  cannot  suppose  that  the  directions  of  vibration 
in  the  incident  and  the  reflected  rays  are  at  right  angles  to  each 
other.  (See,  further,  Chap.  XXXIII.) 

240.  Reflection  and  Refraction  of  Polarised  Light.  —  Young 
first  determined  the  relations  existing  amongst  the  intensities  of  the 
incident,  the  reflected,  and  the  refracted  beams  when  light  falls 
perpendicularly  upon  the  bounding  surface  of  two  transparent 
media. 

Fresnel,  starting  from  certain  assumptions,  gave  a  complete 
investigation  of  these  relations  for  all  angles  of  incidence. 

He  assumed,  first,  the  conservation  of  vis  viva  (or  energy) ;  second, 
continuity  of  displacement  of  the  particles  of  the  ether  at  either 
side  of  the  bounding  surface  ;  third,  proportionality  of  the  density 
of  the  ether  in  a  given  medium  to  the  square  of  the  refractive 
index  of  that  medium. 

The  third  assumption  implies  that  the  rigidity  of  the  ether 
(regarded  as  possessing  properties  analogous  to  those  of  an  elastic 
solid)  is  the  same  in  any  two  media.  For  the  refractive  indices  are 
inversely  as  the  speeds  of  propagation  of  light  in  the  two  media,  and 
therefore  the  densities  are  inversely  as  the  squares  of  the  speeds. 
But  (§  168)  the  squares  of  the  speeds  are  in  direct  proportion  to  the 
ratios  of  the  rigidity  to  the  density  of  each  medium,  from  which 
it  follows  that  the  rigidity  of  the  ether  in  each  must  be  the  same. 

Let  us  suppose  that  light,  polarised  in  a  plane  which  makes  an 
angle  9  with  the  plane  of  incidence,  is  reflected  from  a  transparent 
surface,  and  let  a  be  the  amplitude  of  its  vibrations.  The  resolved 
parts  of  this  parallel,  and  perpendicular,  to  the  plane  of  incidence 
are  a  sin  9  and  a  cos  0,  respectively.  For  shortness,  let  us  denote 
these  by_p  and  q  ;  and  let  p'  and  q'  denote  the  similar  portions  of 
the  amplitude  of  the  reflected  ray,  while  m  and  n  represent  the 
similar  portions  of  the  refracted  ray. 

18 


274  A   MANUAL    OF   PHYSICS. 

The  fact  of  conservation  of  energy  is  expressed  (§  161)  by  the 
equations 

vp  cos  i  (p2  -p'2)  =  y'p'  cos  r  .  m2, (1) 

vp  cos  i  (<z2  —  q'2)  =  v'p'  cos  r  .  n2, (2) 

where  p  and  p'  represent  the  densities  of  the  ether  in  the  media. 
But,  by  Fresnel's  third  assumption,  we  have 

p'_sin2  i_v^ 
p      sin2  r    v''^ 

whence  (1)  and  (2)  become  respectively 

p2— y2=m2tanicotr (3) 

q'<—  q'2=n2  tani  cot  r (4). 

The   continuity   of  the   displacement,   in  the  case   of  vibration 
parallel  to  the  surface,  necessitates  the  condition 

'=n; (5) 


while,  in  the  case  of  the  vibrations  which  take  place  in  the  plane  of 
reflection,  it  necessitates  the  condition 

(P -\-p')  cos  i  =  m  cosr (6). 

Combining  (3)  and  (6),  (4)  and  (5),  we  obtain 

,           sin  ft -r)  ,„, 

q  —  —q— — , .  ,    s (i )•> 

20  cos  i  sin  r  /ox 

*-     ^nlFHT (8)' 

y=-*ssl-  -^ 


%p  cos  i  sin  r 


(10). 


sin  ft+r)cosft-rf  ' 

If  0=0,  so  that  the  incident  light  is  polarised  in  the  plane  of 
incidence,  and  if  the  incidence  is  perpendicular,  (7)  shows  us  that 
the  ratio  of  the  intensity  of  the  reflected,  to  that  of  the  incident, 
light  is  , 

^'2_sin2  (i~r}__  /i  —  r\z_  //i--l\2 


The   same  equation   shows  that,  when  i  =  90°,   the  whole  of  the 
incident  light  is  reflected, 


LIGHT  :     DOUBLE    REFRACTION,    POLARISATION.  275 

Again,  by  (8),  when  0  =  0,  ^  =  0,  we  see  that  the  intensity  of  the 
refracted  light  bears  to  that  of  the  incident  the  ratio 


n*  _(  2?-  V8/    2    \2 
~*\i+r)  ~      +  l/  ' 


Similar  expressions  may  be  obtained  from  (9)  and  (10).     All  these 
results  have  been  verified  experimentally. 

Let  the  plane  of  polarisation  of  the  reflected  light  make  an  angle 
tf>  with  the  plane  of  incidence.     From  (7)  and  (9)  we  get 


-  =  tan  8  . 

q'     qcos(i—r)  cos  (i  —  r) 

Hence  the  plane  of  polarisation  is  rotated  by  reflection  so  as  to  more 
nearly  coincide  with  the  plane  of  incidence.  The  angle  0  is  equal  to  0 
at  perpendicular  incidence  ;  and  it  diminishes  as  i  increases,  until 
when  i+r  =  90°  —  i.e.,  at  the  polarising  angle  —  it  becomes  zero. 
When  i  increases  beyond  this  value,  0  becomes  negative  ;  and,  at 
grazing  incidence,  0=  -9.  So  long  as  0  and  Q  have  the  same  sign, 
the  difference  of  phase  of  the  two  components  of  the  reflected  vibra- 
tion is  zero  :  at  the  polarising  angle  the  difference  changes  suddenly 
to  TT. 

If  $  be  the  angle  which  the  plane  of  polarisation  of  the  refracted 
light  makes  with  the  plane  of  incidence,  (8)  and  (10)  give 


^  =  -  sec  (i  -  r)  =  tan  9  sec  (i  —  r). 


tan 

The  rotation  of  both  the  planes  of  polarisation  may  be  increased 
by  successive  repetitions  of  the  same  process.  Since  sec  (i—r)  = 
sec  (r-i),  refraction  through  a  parallel  plate  gives  tan  1^2  = 
tan  9  sec  2(i-r). 

Equations  (5)  and  (6)  above  express  the  condition  that  there  shall 
be  continuity  of  displacement  of  the  ether  parallel  to  the  bounding 
surface.  No  account  has  been  taken  of  the  displacement  perpen- 
dicular to  the  surface.  If  we  replace  Fresnel's  assumption  of 
uniform  rigidity  by  the  condition  of  no  normal  discontinuity,  we 
find  p  =  p'.  Hence  Fresnel's  third  assumption  is  inconsistent  with 
normal  continuity  of  displacement. 

Making  the  assumption  that  the  ether  is  of  uniform  density  in  all 
media,  Maccullagh  and  Neumann  deduced  expressions  for  the  ampli- 
tudes of  the  components  of  the  vibration  of  the  reflected  ray,  in 
which  the  quantities  on  the  right-hand  sides  of  (7)  and  (9)  are 

18—2 


276  A   MANUAL    OF   PHYSICS. 

simply  interchanged.  Hence,  on  this  theory,  we  must  assume  that 
the  vibrations  are  in  the  plane  of  polarisation.  A  similar  interchange 
occurs  in  the  expressions  for  the  components  of  the  vibration  in  the 
refracted  ray ;  and,  in  addition,  the  magnitudes  are  altered.  But 
this  alteration  occurs  in  such  a  way  that  the  amplitude  of  vibration 
in  the  first  medium,  after  refraction  through  a  parallel  plate,  is 
identical  on  both  theories.  And,  further,  the  rotations  of  the 
planes  of  polarisation  are  of  the  same  magnitude  and  sense  on  the 
two  theories.  Therefore  none  of  the  phenomena  with  which  we  are 
now  dealing  are  capable  of  furnishing  a  test  between  the  assumptions 
of  uniform  density  and  uniform  rigidity. 

241.  Plane,  Circular,  and  Elliptic  Polarisation. — In  the  special 
examples  considered  in  last  section,  the  difference  of  phase  of  the 
two  rectangular  components  of  the  resultant  vibration  was  0  or  ?r. 
But  the  resultant  of  two  rectangular  simple  harmonic  motions  is,  in 
general  (§  52),  elliptic  motion.     Not  only  is  this  true  of  two  rec- 
tangular components ;  it  is  true  of  any  number  of  simple  harmonic 
components  in  lines  inclined  at  any  angles  to  each  other. 

Hence,  if  we  can  assume  that  the  vibrations  of  a  particle  of  the 
ether  are  simply  harmonic  when  polarised  light  (such  as  we  have 
hitherto  considered)  is  passing,  we  must  conclude  that  the  most 
general  vibration  of  such  a  particle,  when  subject  to  various  simul- 
taneous disturbances,  is  elliptical.  This  assumption  is  justified  by 
the  fact  that  no  phenomena  of  light,  which  are  not  due  to  simple 
superposition  of  displacements,  are  observed. 

The  resultant  elliptic  path  is  described  continuously  so  long  as 
the  amplitudes,  phases,  and  periods  of  the  components  remain  con- 
stant. In  this  case  the  light  is  said  to  be  elliptically  polarised. 
As  a  particular  case,  when  all  the  components  can  be  compounded 
into  two  rectangular  components,  equal  in  amplitude  and  period,  but 
differing  in  phase  by  7r/2,  the  ellipse  becomes  a  circle,  and  the  light 
is  circularly  polarised. 

Ordinary  polarisation — for  example,  that  produced  by  reflection — 
occurs  when  the  components  can  be  reduced  to  two  which  differ  in 
phase  by  any  multiple  of  TT.  This  is  usually  termed  plane  polarisa- 
tion, in  order  to  distinguish  it  from  the  above  forms. 

242.  Nature  of  Common  Light. — Common  light  exhibits  no  trace 
of  polarisation  of  any  description.     But  this  is  known  to  be  true 
also  of  plane  polarised  light  if  its  plane  of  polarisation  be  made  to 
rotate  very  rapidly — so  rapidly  that,  in  little  more  than  one-tenth  of 
a  second,  the  directions  of  vibration  have  been  distributed  uniformly 
in  all  possible  orientations  perpendicular  to  the  ray.      Hence  we 
may  conclude  that  ordinary  light  consists  of  elliptically  polarised 


LIGHT  :     DOUBLE    REFRACTION,    POLARISATION.  277 

light,  the  magnitude,  form,  and  position  of  the  ellipse  being  in  a 
constant  state  of  rapid  change. 

But  the  phenomena  of  interference  of  light  show  that  practically 
no  change  occurs  in  the  course  of  some  thousands  of  vibrations,  for 
many  thousands  of  interference  bands  can  be  counted  when  homo- 
geneous light  is  used.  On  the  other  hand,  since  light  travels  at  .the 
rate  of  186,000  miles  per  second,  while  the  length  of  a  wave  is,  on 
the  average,  about  one  forty-thousandth  part  of  an  inch,  many 
millions  of  millions  of  vibrations  must  take  place  per  second.  But, 
again,  as  Stokes  has  pointed  out,  from  the  facts  that  every  source  of 
common  light  consists  of  a  practically  infinite  number  of  points,  and 
that  the  light  emanating  from  each  of  these  points  is,  in  general, 
totally  independent  of  that  issuing  from  any  other  in  respect  of 
direction  of  vibration  and  also  in  respect  of  phase,  we  cannot  expect 
anything  else  than  an  average  effect  in  which  there  is  no  manifesta- 
tion of  '  sides.' 

It  follows  that  a  beam  of  common  light  must  necessarily  be 
divided  into  two  beams  of  equal  intensity  when  it  is  transmitted 
through  a  doubly  refracting  substance. 

From  the  formulae  (7)  and  (9)  of  §  240,  we  see  that  the  total 
intensity  of  the  reflected  portion  of  a  beam  of  unit  intensity  which 
falls  on  a  transparent  substance  is 

I/sin2  (i-r)       tan2  (i  —  r)\ 
2V  sin2  (i+r)  "*~  tan2  (i+r))' 

since  the  two  oppositely  polarised  parts  into  which  the  incident 
beam  is  supposed  to  be  divided  are  of  equal  intensity.  Now  the 
second  term  in  this  expression  is,  in  general,  smaller  than  the  first, 
and  so,  in  the  reflected  beam,  there  is  an  excess  of  light  polarised  in 
the  plane  of  incidence.  The  second  term  vanishes  when  i  +  r  =  90°, 
from  which  it  follows  that  the  reflected  light  is  entirely  polarised  in 
the  plane  of  incidence  at  the  polarising  angle. 

Similarly,  we  can  show  that  the  refracted  beam  contains  an 
excess  of  light  polarised  perpendicularly  to  the  plane  of  incidence, 
that  it  is  entirely  polarised  in  the  perpendicular  plane  at  the  polaris-  ' 
ing  angle,  and  that  its  intensity  is  then  equal  to  that  of  the  re- 
flected beam.  At  all  angles  of  incidence  there  are  equal  amounts 
of  polarised  light  in  the  two  beams. 

The  known  laws  of  the  polarisation  of  common  light  by  reflection 
and  refraction  are  therefore  consequences  of  the  undulatory  theory. 

243.  Metallic  Reflection. — Mains  observed  that  light  is  never 
completely  polarised  by  reflection  from  the  surface  of  metals,  but 


278  A   MANUAL    OF    PHYSICS. 

that  the  polarisation  attained  a  maximum  at  a  certain  angle  of 
incidence.  He  also  observed  that  polarised  light  appeared  to  be 
completely  depolarised  by  reflection  from  a  metallic  surface  when 
its  plane  of  polarisation  was  inclined  at  an  angle  of  45°  to  the  plane 
of  incidence. 

Brewster  verified,  and  extended,  these  results.  He  showed  that 
the  reflected  portion  of  a  beam  of  common  light  might  be  com- 
pletely polarised  by  a  sufficient  number  of  reflections  under  like 
conditions — a  result  previously  inferred  by  Biot.  He  found  also 
that,  when  the  incident  ray  is  polarised  in,  or  perpendicular  to,  the 
plane,  of  incidence,  the  reflected  ray  is  still  polarised  in  the  same 
plane  ;  that,  when  the  original  polarisation  is  in  any  plane  other 
than  these,  partial  depolarisation  seems  to  take  place  ;  and  that  the 
depolarisation  is  greatest  at  the  angle  of  maximum  polarisation. 
Further,  a  second  reflection,  in  the  same  plane,  and  at  the  same 
angle,  repolarises  the  light ;  and  the  new  plane  of  polarisation  lies 
on  the  opposite  side  of  the  plane  of  incidence  and  makes  a  different 
angle  with  it. 

The  '  depolarisation '  above  spoken  of  does  not  mean  restoration 
to  the  condition  of  common  light.  The  originally  polarised  light 
may  be  decomposed  into  two  parts,  one  polarised  in  the  plane  of 
incidence,  the  other  polarised  in  the  perpendicular  plane.  The 
amplitudes  of  these  parts  may  suffer  change  by  reflection,  which 
(§  240)  produces  a  rotation  of  the  plane  of  polarisation.  The  phases 
may  also  be  altered,  and  this  will  give  rise  to  elliptic  polarisation. 
Jamin's  experiments  on  this  subject  show  that  Brewster's  '  depo- 
larisation '  is  really  elliptical  polarisation,  and  also  show  the  nature 
of  the  variations  of  amplitude  and  phase. 

The  laws  of  change  of  amplitude  are  the  same  as  those  already 
found  in  §  240.  The  difference  of  phase  increases  from  perpen- 
dicular incidence  to  grazing  incidence  by  the  total  amount  TT,  the 
phase  of  the  ray  polarised  in  the  plane  of  incidence  being  accelerated 
with  reference  to  that  of  the  other.  The  change  of  phase  is  ex- 
tremely slow,  except  in  the  immediate  neighbourhood  of  the  angle 
of  maximum  polarisation,  between  near  limits  on  either  side  of 
which  the  total  change  occurs. 

The  difference  of  phase  ought,  according  to  Fresnel's  theory 
(§  240),  to  increase  suddenly  by  TT  at  the  polarising  angle. 

Extending  his  observations  to  transparent  bodies,  Jamin  found  that 
the  difference  between  them  and  metals  is  only  a  difference  of  degree. 

In  all  cases  elliptic  polarisation  is  produced,  and  the  maximum 
ellipticity  occurs  at  the  angle  of  maximum  polarisation,  which  co- 
incides very  closely  with  the  angle  deduced  from  Brewster's  law. 


LIGHT  :     DOUBLE    REFRACTION,    POLARISATION.  279 

Jamin  found  that  some  transparent  substances  differ  from  metals 
with  respect  to  the  sign  of  the  difference  of  phase  which  is  produced 
by  reflection.  Substances  whose  refractive  index  is  less  than  1'46 
retard  the  phase  of  the  component  which  is  polarised  in  the  plane 
of  incidence  ;  substances  which  have  a  refractive  index  exceeding 
T46  resemble  metals  in  accelerating  the  phase  of  this  component ; 
substances  in  which  the  refractive  index  is  equal  to  1'46  obey 
Fresnel's  laws. 

He  also  found  that,  in  metals,  the  angle  of  maximum  polarisation 
decreases  as  the  wave-length  of  the  light  increases ;  from  which  we 
see  that  metals,  if  they  obey  Brewster's  law,  must  refract  light  of 
long  wave-length  more  than  light  of  short  wave-length.  Eecent 
experiments  on  refraction  through  thin  metallic  prisms  seem  to 
confirm  this  conclusion. 

When  light  polarised  in  the  plane  of  incidence  is  reflected  from  a 
metallic  surface,  the  intensity  of  the  reflected  beam  is  a  minimum 
at  the  angle  of  maximum  polarisation.  Maccullagh  pointed  out 
that  transparent  substances,  the  refractive  index  of  which  exceeds 
2  +  v3,  possess  (according  to  his  theory)  a  minimum  reflecting 
power  at  a  definite  angle  of  incidence. 

244.  Double  Eefraction   by  Biaxal   Crystals.  —  Brewster   dis- 
covered that  most  doubly  refracting  crystals  possess  two  optic  axes. 
In  uniaxal  crystals  the  axis  is  equally  inclined  to  the  three  edges 
which  meet  at  an  obtuse-angled  corner  of  the  crystal.     In  biaxal 
crystals  the  lines  which  bisect  the  two  angles  contained  by  the  axes, 
and  the  line  at  right  angles  to  these  two,  have  a  definite  relation  to 
the  crystalline  form. 

Fresnel  has  proved,  theoretically  and  experimentally,  that  neither 
of  the  two  rays  in  a  biaxal  crystal  obeys  the  ordinary  law  of  refrac- 
tion. By  means  of  certain  assumptions,  he  investigated  the  problem 
of  the  propagation  of  waves  of  transverse  vibration  in  a  non-isotropic 
elastic  medium.  Crystalline  substances  are  known  to  be  non- 
isotropic,  and  presumably  the  property  is  impressed  upon  the  ether 
which  pervades  them,  and  which  is  known  to  be  hampered,  as  regards 
its  free  oscillation,  by  the  presence  of  material  particles. 

The  complete  laws  of  double  refraction  may  most  readily  be 
studied  from  the  point  of  view  of  his  theory,  which  we  now 
proceed  to  give. 

245.  Fresnel's   Theory  of  Double  Eefraction. — Fresnel  under- 
took his  investigation  when  the  discovery  of  double  refraction  in 
biaxal  crystals  made  it  apparent  that  Huyghens'  construction  for 
the  wave-surface  was  not  applicable  in  all  cases. 

In  a  non-isotropic  substance,  the  resultant  force  which  opposes 


280  A   MANUAL   OF   PHYSICS. 

the  displacement  of  a  particle  does  not  in  general  act  in  the  direction 
of  the  displacement.  But  Fresnel  showed  that  there  are  three 
directions,  at  right  angles  to  each  other,  in  which  the  force,  called 
into  existence  by  the  displacement,  acts  so  as  to  move  the  particle 
directly  back  to  its  position  of  equilibrium.  He  showed  that  if, 
from  any  point  in  the  interior  of  the  substance,  lines  be  drawn  with 
lengths  proportional  to  the  square  roots  of  the  elastic  forces  which 
resist  displacement  in  the  directions  in  which  the  lines  are  taken, 
the  extremities  of  these  lines  will  lie  on  an  ellipsoid  (called  the 
ellipsoid  of  elasticity).  The  three  principal  axes  of  this  surface  are 
in  the  directions  in  which  the  force  tends  to  move  the  displaced 
particle  directly  back  to  its  undisturbed  position. 

The  speed  of  wave-propagation  in  an  elastic  medium  is  propor- 
tional (cf.  §  168)  to  the  square  root  of  the  elastic  force  (or  distor- 
tional  rigidity),  and  hence  the  radii  of  the  ellipsoid  are  proportional 
to  the  speeds  of  propagation  of  waves  when  the  vibrations  are  along 
the  given  radii. 

Consider  a  plane  wave  passing  through  the  medium.  Fresnel 
proved,  from  the  fact  that  the  intensity  of  a  beam  of  light  which  is 
compounded  of  two  beams  polarised  at  right  angles  to  each  other  is 
independent  of  the  phase  of  either  component,  that  the  vibrations 
must  lie  in  the  wave-front.  If,  therefore,  we  regard  a  central 
section  of  the  ellipsoid  of  elasticity  by  the  wave-front,  we  see  that 
there  are  only  two  directions — those  of  the  two  axes  of  the  section — 
in  which  a  displacement  will  give  rise  to  a  reverse  force  acting  in  a 
plane  which  is  normal  to  the  wave  and  which  passes  through  the  line 
of  displacement ;  for  Fresnel  showed  that  the  force  acts  in  the  normal 
to  that  central  section  of  the  ellipsoid  which  is  conjugate  to  the  direc- 
tion of  the  displacement.  In  general,  the  force  will  have  a  compo- 
nent perpendicular  to  the  wave-front ;  but,  according  to  assumption, 
this  produces  no  effect  in  the  way  of  wave-propagation. 

Corresponding  to  any  given  plane  wave-front,  there  are  therefore 
only  two  directions  of  vibration  such  that  the  elastic  force  developed 
by  the  displacement  has  an  effective  component  entirely  in  the 
direction  of  the  displacement.  But  this  condition  is  essential  to  the 
propagation  of  a  permanent  wave.  Hence  a  plane  wave,  incident 
upon  such  a  medium,  is,  in  general,  broken  up  into  two  waves, 
which  are  propagated  in  different  directions  with  speeds  which 
are  proportional  to  the  radii  of  the  ellipsoid  drawn  in  these 
directions. 

[The  following  extract  from  Stokes'  '  Lectures  on  Light '  will  aid 
in  the  formation  of  clear  ideas  on  this  point : 

'  Now  we  have  not  far  to  go  to  find  a  mechanical  illustration  of 


LIGHT  :     DOUBLE    REFRACTION,    POLARISATION.  281 

such  a  mode  of  action.  Imagine  an  elastic  rod  terminated  at  one 
end,  and  extending  indefinitely  in  the  other  direction.  Let  the  rod 
be  rectangular  in  section,  the  sides  of  the  rectangle  being  unequal,  so 
that  the  rod  is  stiffer  to  resist  flexure  in  one  of  its  principal  planes 
than  the  other.  Let  this  rod  be  joined  on  to  a  cylindrical  rod  form- 
ing a  continuation  of  it  which  extends  indefinitely.  Conceive  the 
compound  rod  as  capable  of  propagating  small  transverse  disturb- 
ances, in  which  the  axis  of  the  rod  suffers  flexure.  Imagine  a 
small  disturbance,  suppose  periodic,  to  be  travelling  in  the  cylindrical 
rod  towards  the  junction.  It  will  travel  on  without  change  of  type, 
even  though  the  flexure  of  the  axis  be  not  in  one  plane.  But  to 
find  what  disturbance  it  excites  in  the  rectangular .  rod,  we  must 
resolve  the  disturbance  in  the  cylindrical  rod  into  its  components  in 
the  principal  planes  of  the  rectangular  rod,  and  consider  •  them 
separately.  Each  will  give  rise  in  the  rectangular  rod  to  a  disturb- 
ance in  its  own  plane,  but  the  two  will  travel  along  the  rod  with 
different  velocities.  This  illustrates  the  sub-division  of  a  beam  of 
common  light  falling  on  a  block  of  Iceland  spar  into  two  beams 
polarised  in  rectangular  planes,  which  are  propagated  in  the  spar 
with  different  velocities.  Again,  suppose  the  original  disturbance  in 
the  cylindrical  rod  confined  to  one  plane.  If  this  be  either  of  the 
principal  planes  of  the  rectangular  rod,  the  more  slowly  or  the  more 
quickly  travelling  kind  of  disturbance,  as  the  case  may  be,  will 
alone  be  excited  in  the  latter ;  and  if  the  plane  of  the  original  dis- 
turbance be  any  other,  the  components  into  which  we  must  resolve 
it  in  order  to  find  the  disturbance  excited  in  the  rectangular  rod  will 
in  general  be  of  unequal  intensity,  their  squares  varying  with  the 
azimuth  of  the  plane  of  the  original  disturbance  in  accordance  with 
Malus's  law.  This  illustrates  the  sub-division  of  a  beam  of  polarised 
light  incident  on  Iceland  spar  into  two  of  unequal  intensity  polarised 
in  rectangular  planes,  and  their  alternate  disappearance  at  every 
quarter  of  a  turn.  We  see  with  what  perfect  simplicity  the  theory 
of  transverse  vibrations  falls  in  with  the  elementary  facts  of 
polarisation  discovered  by  Huyghens,  standing  in  marked  contrast 
in  this  respect  with  the  conjecture  by  which  Huyghens  himself 
attempted  to  account  for  double  refraction.'] 

In  any  one  direction  in  the  interior  of  the  substance  two  plane 
waves  may  be  propagated,  generally  with  different  velocities  ;  and 
the  vibrations  in  these  waves  are  necessarily  at  right  angles  to  each 
other.  But  there  are  two  directions  in  which  the  speed  of  propaga- 
tion of  a  wave  is  independent  of  the  direction  of  vibration.  These 
directions  are  parallel  to  the  normals  to  the  two  sets  of  circular 
sections  of  the  ellipsoid  of  elasticity ;  for,  all  the  radii  of  a  circular 


282  A   MANUAL    OF    PHYSICS. 

section  being  equal  in  length,  the  same  force  of  restitution  is  called 
into  play  by  a  given  displacement  along  any  radius. 

The  resolution  of  an  incident  beam  into  two  rectangularly 
polarised  beams  which  obey  Malus's  law,  and  the  existence  of  two 
directions  in  which  a  polarised  beam  is  transmitted  without  modifi- 
cation, are  therefore  consequences  of  Fresnel's  theory. 

The  planes  of  polarisation  pass  through  the  normal  to  the  wave 
and  through  the  major  and  minor  axes  of  the  section  of  the  ellipsoid 
of  elasticity  by  the  wave.  But  the  lines  in  which  the  circular  sec- 
tions cut  the  given  plane  section  are  equally  inclined  to  the  principal 
axes.  Hence  the  planes  of  polarisation  bisect  the  dihedral  angles 
which  are  contained  by  the  planes  which  pass  through  the  normal 
to  the  wave  and  the  optic  axes. 

The  form  of  a  wave  which  spreads  out  through  the  medium  from 
any  centre  is  formed  by  the  following  construction,  which  is  due  to 
Fresnel:  Along  the  normals  to  any  central  plane  section  of  an 
ellipsoid,  similar  to  the  ellipsoid  of  elasticity,  measure  from  the 
centre  lengths  which  are  proportional  to  the  principal  axes  of  the 
section.  The  wave-surface  is  the  locus  of  the  points  so  found,  and 
consists  of  two  sheets.  The  directions  of  the  refracted  rays  are 
found  from  this  surface  by  Huyghens'  construction.  The  tangent 
plane  drawn  to  one  sheet  gives  the  direction  of  the  one  ray ;  that 
drawn  to  the  other  sheet  gives  the  direction  of  the  second  ray. 

The  wave-surface  possesses  symmetry  with  respect  to  the  three 
principal  planes  of  the  ellipsoid  from  which  it  is  derived.  Its  traces 
upon  these  planes  consist  respectively  of  a  circle  and  an  ellipse.  Let 
OA,  OE,  OF  (Fig.  146)  be  the  three  principal  axes.  AB  is  a  circle 
of  radius  OA  equal  to  the  mean  principal  axis  of  the  ellipsoid  ;  CD 
and  EF  are  circles,  the  radii  of  which  are  respectively  equal  to  the 
least  and  the  greatest  principal  axes.  BC  is  an  ellipse,  the  principal 
axes  of  which  are  equal  to  the  two  least  axes  of  the  ellipsoid,  and 
so  on. 

When  the  ellipsoid  of  elasticity  is  one  of  revolution,  the  wave- 
surface  consists  of  two  separate  sheets — a  sphere  and  an  ellipsoid — 
which  have  one  axis  in  common.  Hence  Fresnel's  theory  explains 
the  double  refraction  of  uniaxal  crystals. 

It  is  very  desirable  to  note  that  Fresnel's  results,  although  they 
are  all  seemingly  in  complete  accordance  with  observed  facts,  are 
not  rigorous  deductions  from  his  assumptions.  Green  and  Neumann 
proved  that  a  strict  investigation,  based  upon  these  assumptions, 
will  lead  only  approximately  to  Fresnel's  Laws.  Green  then 
deduced  these  laws  as  rigorous  consequences  of  an  originally  more 
general  theory,  the  generality  of  which  was  subsequently  limited  by 


LIGHT  :     DOUBLE    REFRACTION,    POLARISATION. 


283 


imposing  a  very  probable  condition ;  but  this  theory  made  it  neces- 
sary to  suppose  that  the  direction  of  vibration  is  in  the  plane  of 
polarisation.  He  then  showed  that  a  still  more  general  theory 
would,  by  means  of  suitable  assumptions,  lead  to  the  same  laws, 
and  to  the  conclusion  that  the  vibrations  are  perpendicular  to  the 
plane  of  polarisation.  Other  theories  also  give  like  results;  for 
example,  Maccullagh's  theory,  which  is  identical  in  its  results  with 
Green's  first  theory.  On  this  point  Stokes  remarks  that  the  principle 
of  transverse  vibrations  is  common  to  all  these  theories,  while 
Fresnel's  Laws  are  the  simplest  which  can  suit  the  phenomena ;  so 
that  their  mutual  agreement  can  merely  be  regarded  as  a  confirma- 
tion of  that  principle,  while  none  of  the  special  assumptions  made 
as  to  the  nature  of  the  luminiferous  medium  in  the  interior  of 


FIG.  146. 

crystals  can  be  regarded  as  being  proved  solely  by  the  correctness  of 
the  results  to  which  they  lead. 

246.  Conical  Eefraction. — Sir  W.  E.  Hamilton  showed  that  the 
tangent  plane  QR  (Fig.  146)  touches  the  wave-surface  at  all  points 
of  a  circle  of  contact,  so  that  the  point  P  is  a  '  conical  point.'  Four 
such  points,  one  in  each  of  the  quadrants  in  the  plane  EOA,  exist. 
They  are,  of  course,  the  four  points  of  intersection  of  the  circle 
APB  with  the  eUipse  DPE. 

The  line  OQ  is  perpendicular  to  the  plane  QR.  But  the  perpen- 
dicular on  the  plane  which  touches  the  wave-surface  represents  the 
speed  of  propagation  of  the  plane  wave  to  which  it  is  a  normal ;  and 
the  direction  of  the  rays  in  the  crystal  are  those  of  the  lines  joining 
O  to  the  point  of  contact.  Hence  a  plane  wave,  incident  upon  the 
crystal  in  such  a  direction  that  RQ  is  its  front  after  refraction, 


284  A   MANUAL   OF   PHYSICS. 

gives  rise  to  a  cone  of  rays  which  proceed  in  the  directions  of  the 
lines  joining  0  to  the  various  points  of  the  circle  of  contact. 
Hamilton's  theoretical  prediction  of  this  phenomenon,  which  is 
known  as  internal  conical  refraction,  was  verified  experimentally 
by  Lloyd. 

Both  sheets  of  the  surface  are  touched  by  the  plane  KQ,  and 
hence  there  exists  only  one  wave  velocity  in  the  direction  OQ,  which 
is  therefore  one  of  the  optic  axes.  The  other  optic  axis  is  the  image 
of  OQ  in  the  plane  EOF. 

An  infinite  number  of  tangent  planes  may  be  drawn  to  the  sur- 
face at  the  point  P,  and  therefore  a  ray  which  proceeds  through  the 
crystal  in  the  direction  OP,  gives  rise  on  emergence  to  a  thin  conical 
sheet  of  rays.  This  external  conical  refraction  was  also  predicted 
theoretically  by  Hamilton,  and  found,  as  the  result  of  experiment, 
by  Lloyd. 

Since  the  radii  of  the  wave-surface  represents  the  velocities  of  the 
rays,  we  see  that  OP  is  a  direction  of  single  ray-velocity  in  the  sub- 
stance, for  the  two  sheets  of  the  wave-surface  intersect  at  the  point 
P.  The  other  axis  of  single  ray-velocity  is  the  image  of  OP  in  EOF. 
The  axes  of  single  ray-velocity  never  deviate  much  from  the  axes  of 
single  wave-velocity,  i.e.,  from  the  optic  axes. 

247.  Interference  of  Polarised  Light. — The  laws  of  interference 
of  polarised  light  were  investigated  experimentally  by  Fresnel  and 
Arago. 

Two  rays  of  light,  which  are  polarised  in  perpendicular  planes,  do 
not  give  rise  to  phenomena  of  interference  under  circumstances  in 
which  rays  of  ordinary  light  would  interfere. 

The  planes  of  polarisation  of  two  such  rays  may,  by  suitable 
means,  be  made  to  coincide ;  but  no  interference  will  take  place 
unless  the  two  rays  were  originally  parts  of  one  polarised 
beam. 

Eays  polarised  in  the  same  plane  will  always  interfere  under  the 
conditions  in  which  rays  of  ordinary  light  would  interfere. 

When  rays  which  have  been  polarised  by  double  refraction  pro- 
duce interference,  the  phenomena  exhibited  are  such  as  to  necessitate 
the  assumption  that  the  phase  of  one  component  has  been  accele- 
rated by  the  amount  TT  relatively  to  that  of  the  other.  The  reason 
for  this  is  not  far  to  seek. 

Let  OP  (Fig.  147)  represent  the  vibration  in  a  ray  which  falls  upon 
a  crystal  of  Iceland  spar  the  principal  section  of  which  is  AA'.  OP 
will  be  broken  up  into  two  components  On  and  Om,  respectively 
along  and  perpendicular  to  the  axis.  Let  each  of  these  be  again 
resolved  in  the  directions  aa'  and  bb'.  The  components  along  aa' 


LIGHT  :     DOUBLE    REFRACTION,    POLARISATION.  285 

are  in  the  same  phase,  but  the  components  along  W  are  necessarily 
in  opposite  phases. 

248.  Colours  of  Crystalline  Plates. — Let  us  suppose  that  a  beam 
of  plane  polarised  white  light  is  obtained  by  means  of  reflection, 
and  let  a  second  reflector  be  so  placed  as  to  extinguish  the  beam. 
If  a  thin  crystalline  plate  be  now  interposed  between  the  two 
reflectors  in  the  path  of  the  beam,  intensely  coloured  light  will  in 
general  be  reflected  from  the  second  surface. 

The  light  disappears  whenever  the  principal  section  of  the 
crystalline  plate  coincides  with,  or  is  perpendicular  to,  the  plane  of 
reflection  from  the  first  surface.  If  the  plate  be  turned  round,  in 
its  own  plane,  from  this  position,  some  reflection  from  the  second 
surface  will  be  evident,  and  the  reflected  light  will  vary  in  intensity, 


FIG.  147. 

but  not  in  colour,  as  the  plate  is  rotated.  (The  same  appearances, 
colour  excepted,  would,  of  course,  be  manifested  even  if  the  plate 
were  thick.)  On  the  other  hand,  if  the  plane  of  reflection  at  the 
second  surface  be  varied,  the  crystal  being  fixed,  the  colour  changes 
gradually  into  the  tint  which  is  complementary  to  the  former. 
When  any  two  successive  positions  of  the  second  plane  of  incidence 
differ  by  90°  from  each  other,  the  reflected  tints  are  complemen- 
tary. 

The  colours  depend  upon  the  thickness  of  the  crystalline  plate, 
and  vary  with  the  thickness  in  the  same  way  as  the  colours  seen  by 
reflection  from  a  thin  plate  of  air. 

The  explanation  of  the  phenomenon  is  simple.  The  originally 
plane  polarised  light  is  divided  in  the  crystal  into  two  beams,  which 
are  oppositely  polarised,  and  which  traverse  the  plate  with  unequal 
speeds.  The  phase  of  the  one  beam  is  therefore  retarded  relatively 


286 


A   MANUAL    OF    PHYSICS. 


to  that  of  the  other,  and  so  interference  may  take  place  if  the  vibra- 
tions in  the  two  portions  are  again  resolved  in  a  common  direction. 
The  tint  is  due  to  the  cutting  out  of  some  rays,  and  the  intensify- 
ing of  others,  by  interference. 

The  light  which  emerges  from  the  plate  is  elliptically  polarised, 
since  it  is  compounded  of  two  rectangularly  polarised  beams  which 
differ  from  each  other  in  phase.  In  particular,  when  the  difference 
of  phase  is  any  odd  multiple  of  a  quarter  of  a  period,  the  light 
is  circularly  polarised ;  and,  when  the  difference  is  a  multiple  of 
half  of  a  period,  the  light  is  plane  polarised.  If  the  difference  is  an 
even  multiple  of  a  half-period,  the  plane  of  polarisation  coincides 
with  the  original  plane ;  it  makes  an  equal  angle  with  the  principal 
section,  on  the  opposite  side,  when  the  difference  is  an  odd  multiple 
of  a  half-period. 


FIG.  148. 


Let  POP'  (Fig.  148)  represent  the  direction  of  vibration  in  the 
incident  beam  of  light  which  falls  upon  the  plane  surface  P  0  M  of 
a  doubly  refracting  plate,  and  let  p,  //  represent  the  principal  section 
of  the  crystal  so  that  the  vibrations  in  the  ordinary  ray  are  in  the 
direction  0  M.  If  the  amplitude  of  vibration  along  0  P  be  unity, 
the  amplitude  of  the  vibrations  in  the  ordinary  ray  is  cos  9,  where 
9  =  P  0  M ;  and  the  amplitude  of  those  in  the  extraordinary  ray  is 
sin  9.  Let  v  v'  represent  the  principal  plane  of  a  second  doubly 
refracting  crystal  through  which  pass  both  the  beams  into  which  the 
original  beam  is  divided. 

The  vibrations  along  0  M  and  0  ^  give  rise  to  two  sets  of  vibra- 
tions along  0  N,  the  amplitudes  of  which  are  cos  9  cos  (0  —  0)  and 
sin  9  sin  (0—0)  respectively,  where  0  =  P  O  N.  These  two  vibra- 
tions are  the  components  of  the  vibrations  of  the  ordinary  ray  which 
emerges  from  the  second  crystal. 


LIGHT  1     DOUBLE    REFRACTION,    POLARISATION.  287 

The  vibrations  along  0  M  and  0  p  also  give  rise  to  two  sets  of 
vibrations  along  0  v,  the  amplitudes  of  which  are  cos  9  sin  (0  —  0) 
and  sin  9  cos  (9  -  0)  respectively.  And  these  two  vibrations  are  the 
components  of  the  vibrations  of  the  extraordinary  ray  which  emerges 
from  the  second  crystal. 

The  intensity  of  the  ordinary  beam  is  therefore  cos2  9  cos2  (9  -  0) 

+  sin2  9  sin2  (9-0)  +2  cos  9  sin  0  cos  (0-0)  sin  (0-0)  cos  2  TT_' 

where  I  is  the  effective  difference  of  path  of  the  two  rays  in  the  thin 
crystalline  plate,  and  X  is  the  wave-length.  Similarly,  the  intensity 
of  the  extraordinary  beam  is  cos2  9  sin2  (9  -  0)  +  sin2  9  cos2  (0  -  0) 

+  2  cos  9  sin  9  cos  (0-0)  sin  (0-0)  cos  2  vL 

\ 

These  two  expressions  easily  reduce  to 

cos2  0  -  sin  2  0  sin  2  (0—  0)  sin2  TT  _  , 

/Y 

and  sin2  0  +  sin  2  0  sin  2  (0  -  0)  sin2  TT-  • 

X 

When  light  of  various  wave  lengths  is  used,  the  sum  of  all  quan. 
tities  of  the  form  sin2  TT  lj\  must  be  taken. 

From  these  expressions  we  can  deduce  all  the  observed  effects. 

The  sum  of  the  two  intensities  is  unity,  and  therefore  the  colours 
of  the  ordinary  and  extraordinary  beams  are  complementary. 

The  second  term  in  each  expression  is  the  quantity  upon  which 
the  coloration  depends.  Hence  all  colour  vanishes  when 

0  =  0  or  |, 

that  is,  when  the  principal  section  of  the  crystalline  plate  is  parallel 
or  perpendicular  to  the  original  plane  of  polarisation.  It  also 
vanishes  when 

0-0  =  0  or-, 

that  is,  when  the  principal  sections  of  the  thin  plate  and  the 
second  doubly  refracting  crystal  are  parallel  or  mutually  perpen- 
dicular. The  reason  for  this  is  that,  in  each  of  these  four  cases, 
one  of  the  components  of  both  the  extraordinary  and  the  ordinary 
beams  necessarily  vanishes :  and  the  reason  for  the  colours  of 
the  two  beams  being  complementary  is  that  the  difference  of  the 
phases  of  the  components  of  the  extraordinary  beam  necessarily 


A   MANUAL   OF   PHYSICS. 


differs  from  that  of  the  components  of  the  ordinary  beam  by  the 
amount  ir. 

The  coloration  is  greatest  when  sin  2  9  sin  2  (9  —  0)  is  a  maximum. 
As  the  maximum  value  is  unity,  this  necessitates 


9  = 


=  0  or 


that  is,  the  principal  section  of  the  crystalline  plate  must  make  an 
angle  of  45°  with  the  original  plane  of  polarisation,  and  the  prin- 
cipal section  of  the  second  doubly  refracting  crystal  must  be  parallel, 
or  perpendicular,  to  that  plane. 

The  colours  change  through  regularly  recurring  cycles  as  the 
quantity  Z,  and  therefore  as  the  thickness  of  the  plate,  increases  by 
successive  equal  stages. 

249.  Special  Cases. — Hitherto  we  have  assumed  that  the  incident 
beam  of  light  is  parallel.  We  shall  now  suppose  that  a  diverging 
beam  of  polarised  light  traverses  a  uniaxal  crystalline  plate  the 
parallel  plane  faces  of  which  are  perpendicular  to  the  axis. 

A  ray  O  P  (Fig.  149)  which  passes  perpendicularly  through  the  plate 
suffers  no  change.  It  will  pass  through,  or  be  stopped  by,  a  second 
plate  which  is  cut  parallel  to  its  optic  axis  according  as  the  principal 


FIG.  149. 


section  of  the  second  plate  is  parallel  to,  or  perpendicular  to,  the 
original  plane  of  polarisation.  Any  other  ray  will  undergo  change 
according  to  its  inclination  to  the  optic  axis,  and  according  to  the 
angle  which  the  plane  passing  through  it  and  the  axis  makes  with 
the  plane  of  polarisation. 

Let  A  B'  A'  B  (Fig.  150)  be  perpendicular  to  the  axis  of  the  cone 
of  rays,  and  let  A  A'  and  B  B'  represent  respectively  the  original 
plane  of  polarisation  and  the  perpendicular  plane.  All  rays  which 


LIGHT  :     DOUBLE    REFRACTION,    POLARISATION.  289 

emerge  from  the  crystal  in  these  planes  will  be  allowed  to  pass 
through,  or  will  be  stopped  by,  the  second  plate,  according  as  its 
principal  plane  is  parallel,  or  is  perpendicular,  to  the  original  plane 
of  polarisation.  The  field  will  therefore  exhibit  a  light  or  a  dark 
cross. 

(Haidinger's  Brushes  are  observed  when  polarised  light  is  ex- 
amined by  the  naked  eye.  The  phenomenon  consists  of  two  yellowish- 
brown  patches  of  light  forming  a  brush  the  axis  of  which  is  parallel 
to  the  plane  of  polarisation  ;  and  two  other  bluish  or  purplish  patches 
occur  in  the  angles  between  the  yellow  patches.  The  appearance  is 
due  to  a  polarising  structure  which  is  most  highly  developed  in  dark 
eyes.  It  appears  that  the  yellow  spots  of  the  eye  are  doubly  re- 
fracting and  absorb  the  extraordinary  ray  to  a  greater  extent  than 
the  ordinary  ray.  Helmholtz  finds  that  the  effect  only  appears  with 
blue  light.  The  brushes  soon  disappear  unless  the  plane  of  polarisa- 
tion be  changed  at  intervals.) 

At  any  other  point,  such  as  Q,  the  vibration  of  the  ray  is  resolved 
into  its  two  components,  polarised  parallel  and  perpendicular  to  the 
plane  through  PQ.  Thus  the  incident  ray  will  be  divided  into  two, 
which  traverse  the  crystal  with  different  speeds  and  so  give  rise  to 
interference.  The  retardation  of  the  phase  of  the*  one  component 
relatively  to  that  of  the  other  is  constant  so  long  as  the  distance 
PQ  is  constant,  and  becomes  greater  and  greater  as  PQ  increases 
in  length.  Hence  the  field  exhibits  a  series  of  alternately  light  and 
dark  circles  surrounding  the  point  P. 

The  circles  are  brilliantly  coloured  if  white  light  be  used ;  and  the 


FIG.  151. 

colours  seen  when  the  principal  section  of  the  second  crystal  occu- 
pies any  definite  position  are  exactly  complementary  to  those  which 
are  seen  when  this  plane  is  rotated  through  a  right  angle. 

19 


290 


A    MANUAL    OF    PHYSICS. 


These  effects  are  produced  in  comparatively  thick  crystals,  since 
the  difference  between  the  speeds  of  the  two  rays,  in  directions  not 
greatly  different  from  the  optic  axis,  is  comparatively  small. 

The  squares  of  the  radii  of  successive  circles  are  nearly  propor- 
tional to  the  natural  numbers.  For  it  has  been  proved  that  the 
difference  of  the  squares  of  the  speeds  of  propagation  of  the  two 
waves  is  proportional  to  the  square  of  the  sine  of  the  angle  which 
the  ray  within  the  crystal  makes  with  the  optic  axis,  and  also  that 
it  is  proportional  to  the  thickness  of  the  plate,  and  to  the  interval  of 
retardation  conjointly.  Hence  the  retardation  varies  as  the  square 
of  the  sine  of-  the  angle  between  the  ray  and  the  axis.  But  this 
angle  is  very  nearly  equal  to  the  angle  QOP  (Fig.  149),  or  to  the 
distance  QP. 

Consider  now  a  parallel  plate  cut  from  a  biaxal  crystal  in  a  direc- 


FIG.  152. 


tion  perpendicular  to  the  line  bisecting  the  optic  axes.  The  interva 
of  retardation  is,  in  this  case,  proportional  to  the  product  of  the 
sines  of  the  angles  which  the  wave  normal  makes  with  the  axes. 
And  these  sines  are  approximately  proportional  to  the  distances  of 
the  point  of  emergence  of  the  ray  from  the  points  in  which  lines 
drawn  parallel  to  the  axes,  from  the  point  of  incidence  on  the  first 
face  of  the  crystal,  meet  the  second  face.  Hence  the  locus  of  P  is 
an  oval  of  Cassini. 


LIGHT  :     DOUBLE    REFRACTIONj    POLARISATION.  291 

250.  Artificial  Production  of  the  Doubly  Refracting  Structure. 
— Fresnel  showed  that  glass  and  other  singly  refracting  substances 
become  doubly  refracting  when  subjected  to  stress ;  and  Brewster 
showed  that  unequal  heating  of  the  substance  is  sufficient  to  pro- 
duce the  requisite  strain.      Compression  gives  rise  to  a  structure 
which  resembles  that  of  Iceland  spar  in  producing  an  extraordinary 
ray  the  refractive  index  of  which  is  less  than  that  of  the  ordinary 
ray.     Expansion  produces  an  opposite  effect. 

The  optic  axis,  in  these  cases,  is  fixed  in  position  as  well  as  in 
direction,  unless  the  strain  be  homogeneous. 

If  an  elliptic  cylinder  of  glass  be  suddenly  heated  uniformly  over 
its  surface,  a  biaxal  refracting  structure  is  developed.  This  structure 
is  also  usually  seen  in  unannealed  glass. 

Analogous  changes  may  be  produced  in  substances  which  naturally 
exhibit  double  refraction. 

Clerk-Maxwell  showed  that  a  viscous  liquid  becomes  doubly 
refractive  while  subjected  to  shearing  stress. 

251.  Eotatory  Polarisation. — Quartz  is  a  doubly  refracting  sub- 
stance in  which  the  refractive  index  of  the  extraordinary  ray  is 
greater  than  that  of  the  ordinary  ray.     The  wave-surface  consists  of 
a  sphere  and  a  spheroid,  but  the  spheroid  lies  entirely  within  the 
sphere.     It  follows  from  this  that  the  speeds  of  propagation  of  the 
two  rays  along  the  optic  axis  are  not  the  same.     But,  further,  the 
vibrations  in  the  two  rays  are  not  rectilinear.      They  are  elliptical 
in  general,  and  the  ellipses  are  described  in  opposite  directions  in 
the  two  rays.    When  the  rays  are  propagated  along  the  axis,  the 
vibrations  become  circular. 

Now  the  resultant  of  two  uniform  motions,  of  equal  period,  in 
opposite  directions  in  the  same  circle,  is  rectilinear  motion.  For,  if 
A,  A'  (Fig.  153)  represent  simultaneous  positions  of  the  oppositely 
moving  points,  the  resolved  parts  of  their  motion  perpendicular  to 
the  line  PQ,  which  is  drawn  from  the  centre  of  the  circle  through 
the  middle  point  of  the  arc  joining  A,  A',  destroy  each  other. 
Hence  the  resultant  is  simple  harmonic  motion  in  the  line  PQ. 

But  if  A'  be  retarded  relatively  to  A,  the  line  PQ  will  take  a 
new  position,  bisecting  the  arc  between  A  and  the  new  position 
of  A'.  And,  if  A'  is  continuously  retarded,  PQ  will  revolve  con- 
tinuously round  in  the  direction  A  A'.  On  the  other  hand,  if  A 
be  retarded  relative  to  A',  PQ  will  revolve  in  the  direction  from 
A'  to  A. 

The  two  circularly  polarised  rays  in  quartz  therefore  produce 
plane  polarised  light  on  emergence ;  but,  because  of  the  retardation 
of  one  ray  relatively  to  the  other,  the  plane  of  polarisation  has  been 

19—2 


292 


A   MANUAL    OF    PHYSICS. 


rotated  through  an  amount  which  is  proportional  to  the  thickness  of 
the  quartz. 

The  rotation  is  right-handed  in  some  specimens  of  quartz ;  left- 
handed  in  others.  Amethyst  consists  of  alternate  layers  of  right- 
handed  and  left-handed  quartz. 


The  amount  of  rotation  is  dependent  upon  the  wave-length. 
Biot,  and  subsequently  Broch,  proved  that  it  is  approximately 
inversely  proportional  to  the  square  of  the  wave-length.  The  first 
three  terms  of  the  formula 


furnish  a  much  better  approximation,  p  being  the  rotation,  and  a,  6, 
c  being  constants. 

The  spectrum  produced  by  plane  polarised  sunlight  which  has 
passed  through  a  plate  of  quartz  is  indistinguishable  from  that  which 
is  produced  by  ordinary  sunlight.  But  a  profound  modification 
takes  place  if,  previous  to  its  passage  through  the  refracting  prism, 
the  light  be  passed  through  an  apparatus  arranged  to  polarise  light 
in  a  plane  at  right  angles  to  the  original  plane  of  polarisation. 
Were  the  quartz  plate  taken  away,  no  light  could  pass  under  these 
circumstances.  The  effect  of  the  quartz  is  to  restore  the  light,  with 
the  exception  of  such  rays  as  have  had  their  plane  of  polarisation 
rotated  through  a  multiple  of  two  right  angles.  Consequently,  the 
spectrum  is  crossed  by  dark  bands.  By  rotating  the  second 
polarising  apparatus  any  one  of  the  dark  bands  may  be  caused  to 
occupy  any  desired  position  in  the  spectrum,  and  so  the  total  rota- 
tion for  any  particular  kind  of  light  (and,  consequently,  the  wave- 
length of  that  light)  may  be  measured  with  extreme  accuracy. 


LIGHT  I     DOUBLE    REFRACTION,    POLARISATION.  293 

The  dark  bands  are  not  sharply  marked,  for  rays  near  those  which 
are  totally  extinguished  are  necessarily  partially  extinguished. 
And,  since  portions  of  the  light  are  cut  out,  the  finally  emergent 
light  is  coloured.  The  coloration  disappears  when  the  length  of 
the  quartz  is  so  great  that  the  portions  which  are  cut  out  are  dis- 
tributed with  practical  uniformity  throughout  the  spectrum. 

Many  liquids,  solutions,  and  even  vapours,  possess  this  rotatory 
power — the  rotation  being  in  some  cases  left-handed,  in  others  right- 
handed.  As  a  rule,  the  rotation  produced  by  a  given  thickness  of 
a  liquid  is  much  smaller  than  that  produced  by  the  same  thickness 
of  quartz. 

Neither  dilution  by  an  inactive  substance  nor  vaporisation  alters 
the  rotatory  power  of -a  liquid,  except  in  degree  ;  on  the  other  hand, 
Herschel  showed  that  quartz  is  inactive  when  in  solution ;  and 
Brewster  showed  that  it  is  inactive  when  fused.  Hence  it  is  inferred 
that,  while  the  rotation  in  quartz  depends  upon  the  crystalline 
structure,  the  rotation  in  liquids  and  vapours  is  a  molecular 
phenomenon. 

All  rotatory  polariscopes,  or  saccharimeters,  depend  essentially 
upon  the  above  principles.  In  some  of  them,  equal  intensity  of  two 
beams  of  light  is  used  as  a  test ;  in  others,  the  test  is  furnished  by 
the  equality  of  tint  of  two  beams.  A  spectroscopic  test,  such  as  that 
which  has  been  described  above,  is  by  far  the  most  delicate. 

If  the  light  which  has  passed  through  quartz  be  reflected  back,  so 
as  to  retraverse  it  in  the  opposite  direction,  the  rotation  will  be  un- 
done. Kotation  of  the  plane  of  polarisation  may  be  produced  in  a 
magnetic  field ;  but,  in  this  case,  reversal  of  the  path  of  the  light 
will  not  (Chap.  XXXII.)  be  accompanied  by  an  undoing  of  the  rotation. 

252.  Polarising  Prisms. — Many  methods  are  used  for  the  pro- 
duction of  polarised  light. 

Plane  polarisation  by  reflection  has  been  already  described. 

Any  doubly  refracting  crystal,  of  course,  furnishes  us  with  the 
means  of  producing  plane  polarised  light,  provided  that  we  can 
separate  the  two  beams. 

Some  substances,  such  as  tourmaline,  strongly  absorb  one  of  the 
two  rays  into  which  the  incident  beam  is  divided,  and  so  furnish, 
when  of  sufficient  thickness,  a  ready  means  of  obtaining  plane 
polarised  light. 

A  block  of  Iceland  spar  separates  the  rays  in  proportion  to  its 
length,  but  it  is  extremely  difficult  to  obtain  large  blocks  which  are 
fr/e  from  internal  flaws.  In  Nicol's  prism,  therefore,  one  of  the 
two  rays  is  got  rid  of  by  total  reflection.  A  long  block  of  the  spar 
is  divided  into  two  parts  by  a  plane  which  is  perpendicular  to  its 


294  A   MANUAL    OF   PHYSICS. 

principal  section.  The  two  portions  are  then  cemented  together,  in 
their  original  position,  by  means  of  Canada  balsam,  the  refractive 
index  of  which  is  intermediate  in  magnitude  between  the  indices  of 
the  spar  for  the  extraordinary  and  the  ordinary  rays.  Consequently, 
when  the  inclination  of  the  dividing  plane  to  the  path  of  the  ray  is 
sufficiently  great,  the  ordinary  ray  suffers  total  reflection  at  the  sur- 
face of  the  film  of  balsam,  and  the  extraordinary  ray  alone  is 
transmitted. 

FoucaulVs  prism  is  essentially  similar,  but  the  film  of  balsam  is 
replaced  by  a  film  of  air.  A  shorter  block  of  the  spar  suffices  in 
this  method,  but  considerable  loss  of  light  takes  place  by  reflection 
at  the  surface  of  the  film. 

Considerable  separation  of  the  rays  may  be  obtained  by -the  use  of 
a  prism  of  Iceland  spar,  or  of  quartz,  which  is  achromatised  by 
means  of  a  prism  of  glass.  In  EocJion's  prism  the  edge,  and  one 
of  the  faces,  of  the  prism  are  perpendicular  to  the  optic  axis.  The 
rays  therefore  pass  through  the  prism  without  separation  until  they 
emerge  at  the  opposite  face.  The  prism  is  achromatised  by  means 
of  a  second  prism,  of  the  same  substance,  the  refracting  edge  of 
which  is  parallel  to  the  axis.  The  ordinary  ray  proceeds  through 
the  second  prism  without  alteration  of  direction,  and  is  therefore 
uncoloured  ;  but  the  extraordinary  ray  is  considerably  refracted,  and 
is  coloured  at  its  edges,  since  the  refraction  depends  upon  the  wave- 
length. 

Greater  angular  separation  of  the  rays  may  be  obtained  by 
Wollastori }s  prism.  In  this  arrangement  the  refracting  edge  of  the 
first  prism  is  perpendicular  to  the  optic  axis,  but  the  face  upon  which 
the  light  falls  perpendicularly  is  parallel  to  the  axis.  In  all  other 
respects  the  arrangement  is  the  same  as  in  Eochon's  prism.  Both 
rays  are  coloured  at  their  edges,  for  both  are  deviated  from  their 
original  direction. 

Methods  of  producing  elliptically,  or  circularly,  polarised  light 
have  been  already  indicated.  They  all  depend  upon  the  introduction 
of  a  difference  of  phase  between  the  two  rectangularly  polarised  com- 
ponents of  a  beam  of  light.  This  difference  may  be  produced  by 
transmission  through  a  doubly  refracting  plate.  When  the  plate  is 
of  such  thickness  as  to  produce  a  difference  of  phase  of  a  quarter- 
period,  the  light  is  circularly  polarised  provided  that  the  two  beams 
are  of  equal  intensity.  Such  a  plate  is  termed  a  quarter-wave 
plate.  The  difference  of  phase  may  also  be  produced  by  reflection 
or  refraction — as  in  FresneVs  rhomb.  This  consists  of  a  parallelepiped 
of  St.  Gobain  glass  the  faces  of  which  are  so  inclined  that  a  ray 
enters  and  emerges  perpendicularly  at  opposite  faces  after  two 


LIGHT  :     DOUBLE    REFRACTION,    POLARISATION.  295 

internal  reflections  at  an  angle  of  54°  37'.  Each  reflection  produces 
a  difference  of  phase  of  45°,  and  therefore  the  light  emerges  circu- 
larly polarised  if  it  were  originally  polarised  in  a  plane  inclined  at  45° 
to  the  plane  of  internal  reflection. 

Conversely,  Fresnel's  rhomb  shows  the  existence  of  circularly 
polarised  light  by  changing  it  into  plane  polarised  light.  This  will 
also  occur  in  the  case  of  elliptically  polarised  light  when  either  axis  of 
the  ellipse  is  inclined  at  45°  to  the  plane  of  internal  reflection.  But 
the  two  cases  may  be  distinguished  by  means  of  a  Nicol's  prism ; 
for  rotation  of  the  Nicol  produces  no  alteration  of  intensity  if  the 
beam  which  is  passing  through  it  is  circularly  polarised,  but  it  does 
produce  variation  of  intensity  if  the  beam  is  elliptically  polarised. 
Though  similar  variation  of  intensity  takes  place  if  the  incident 
beam  is  partially  plane  polarised,  a  quarter-wave  plate,  or  a  Fresnel's 
rhomb,  enables  us  clearly  to  distinguish  between  these  two  cases. 


CHAPTEE  XX. 

THE    NATURE    OF   HEAT. 

253.  Radiant  Heat.  Its  Identity  with  Light. — We  are  accustomed 
to  speak  of  the  heat  which  we  receive  from  the  sun  just  as  we  speak 
of  the  light  which  we  receive  from  it.  And  so  the  term  '  radiant 
heat,'  as  applied  to  the  heat  which  comes  to  us  from  distant  bodies, 
apparently  without  the  intervention  of  ordinary  matter,  has  come 
into  scientific  use. 

The  non-intervention  of  ordinary  matter  in  the  process  of  radia- 
tion by  which  heat,  like  light,  passes  from  one  body  to  another  at  a 
distance  from  it,  can  readily  be  proved.  There  is  no  sufficient 
amount  of  ordinary  matter  in  interstellar  or  interplanetary  space  to 
account  for  the  transference  :  and  a  hot  body  cools  in  vacuo  almost 
as  readily  as  when  surrounded  by  air — indeed,  in  certain  cases,  a  hot 
body  wih1  cool  less  rapidly  when  surrounded  by  a  material  medium 
than  it  will  otherwise. 

Luminous  bodies  (those  which  exhibit  fluorescence  or  phosphor- 
escence being  alone  excepted)  radiate  heat  as  well  as  light,  and  the 
hotter  they  are  the  more  luminous  do  they  appear.  We  might, 
therefore,  naturally  conclude  that  the  difference  between  light  and 
radiant  heat  is  not  a  difference  of  kind,  but  only  a  difference  of 
degree.  This  inference  is  fully  borne  out  by  many  facts. 

One  point  in  which  they  completely  resemble  each  other  is  recti- 
linear propagation  in  free  space  or  in  a  homogeneous  medium.  The 
heat-shadow  which  any  obstacle  casts  is  identical  with  the  shadow 
which  it  produces  with  regard  to  light  proceeding  from  the  same 
source.  And,  since  the  path  of  light  is  rectilinear,  this  proves  that 
radiant  heat  also  moves  in  straight  lines. 

Heat,  like  light,  is  not  propagated  instantaneously ;  and  an,  even 
more  fundamental  point  of  resemblance  than  the  above  appears  in 
the  fact  that  their  speeds  in  vacuo  are  identical.  This  is  made 
evident  by  the  simultaneous  disappearance  and  re-appearance  of 
the  light  and  heat  when  a  total  eclipse  of  the  sun  takes  place. 


THE    NATURE    OF    HEAT.  297 

The  laws  of  reflection  of  the  two  are  identical,  for  the  focus  of  a 
mirror  for  heat-rays  is  the  same  as  its  focus  for  light-rays.  A 
thermo-electric  pile,  placed  at  the  focus  of  a  reflecting  telescope,  can 
make  evident  the  heat  radiated  from  a  star. 

Their  laws  of  refraction  are  also  the  same ;  although,  at  first 
sight,  a  difference  appears  because  the  focus  of  a  lens  for  heat  is 
farther  off  from  the  lens  than  its  focus  for  light.  But  this  only 
strengthens  the  analogy,  for,  unless  the  lens  be  achromatised,  the 
focus  for  red  rays  is  farther  off  than  the  focus  for  blue  rays. 

Both  are  governed  by  the  same  laws  of  interference  and  of 
polarisation.  The  phenomena  of  interference  prove  the  existence  of 
periodicity  of  motion,  and  show  that  the  vibrations  are  transverse, 
as  in  the  case  of  light.  And  the  usual  methods  based  upon  interfer- 
ence, diffraction,  and  refraction,  enable  us  to  measure  the  wave- 
length, which  is  found  to  be  greater  than  that  of  luminous  rays. 

We  therefore  conclude  that  light  and  radiant  heat  are  one  and  the 
same  thing ;  that  the  latter  differs  from  the  former  only  as  red  light 
differs  from  blue  light ;  and  that  it  is  not  evident  to  the  sense  of  sight 
because  the  eye  is  so  constituted  that  it  cannot  respond  to  the  slower 
vibrations.  We  already  know  that  some  eyes  can  perceive  rays  at 
the  red  end  of  the  spectrum  (and  also  at  the  blue  end)  which  are 
totally  invisible  to  other  eyes. 

And  colours — the  word  being  used  by  analogy — appear  in  heat 
just  as  they  do  in  light.  Rock-salt  is  very  transparent  to  heat  rays 
just  as  glass  is  transparent  to  light ;  but,  on  the  other  hand,  ordinary 
glass  is  very  opaque  to  heat  rays,  i.e.,  it  absorbs  them  to  a  great 
extent.  It  acts  to  heat  just  as  coloured  glass  acts  to  light ;  and 
many  other  substances  act  similarly.  The  law  of  absorption,  with 
varying  thickness  of  the  medium,  is  the  same  as  that  which  holds  in 
the  case  of  light  (§  205). 

254.  Heat  in  Material  Bodies.  Hypothesis  of  Molecular  Vor- 
tices.— Since  hot  bodies  give  out  radiation,  and  since  the  propagation 
of  radiation  involves  motion  of  the  particles  of  an  inert  medium,  we 
might  infer  that  the  particles  of  a  hot  body  must  be  in  rapid  motion, 
and  that  the  communication  of  heat  from  one  body  to  another 
depends  upon  the  intercommunication  of  motion. 

It  is  scarcely  a  century  since  Rumford  and  Davy  arrived  at  this 
result  upon  experimental  grounds. 

Previous  to  their  investigations  heat  was  supposed  to  be  a  form  of 
matter  which  was  occluded  in  the  interior  of  substances,  but  which 
was  looked  on  as  imponderable  since  it  did  not  add  to  the  weight  of 
a  body.  This  imponderable  substance  was  termed  Caloric. 

According  to  the  caloric  hypothesis,  a  body  was  hotter  or  colder 


298  A   MANUAL    OF    PHYSICS. 

in  virtue  of  its  having  absorbed  a  greater  or  a  smaller  quantity  of 
caloric ;  and  when,  by  any  means,  the  capacity  of  a  body  for  caloric 
was  diminished,  it  gave  out  heat  (or  rather  caloric). 

In  the  year  1798  Rumford  was  engaged  in  the  boring  of  cannon, 
and  observed  (what  had  often  previously  been  noticed]  that  there 
was  a  rise  of  temperature  in  the  process  of  the  reduction  of  the  solid 
metal  to  the  state  of  filings.  But,  according  to  the  caloric  hypothesis, 
the  rise  of  temperature  implies  a  diminution  of  the  capacity  of  the 
substance  for  caloric  ;  and,  conversely,  an  increase  of  the  capacity 
of  a  body  for  caloric  would  be  accompanied  by  a  fall  of  tempera- 
ture unless  additional  caloric  were  supplied.  Hence,  in  Eumford's 
experiments,  the  rise  of  temperature  of  the  filings  signifies,  on  this 
hypothesis,  a  diminution  of  the  capacity  for  caloric. 

Bumford,  therefore,  sought  to  determine  by  experiment  whether 
or  not  the  filings  had  less  capacity  for  heat  than  the  solid  metal 
had.  He  heated  equal  weights  of  the  solid  metal  and  of  the  filings 
to  the  same  high  temperature,  and  dropped  them  into  equal  quan- 
tities of  water  at  the  same  low  temperature.  He  found  that  the 
same  changes  of  temperature  were  produced  in  both  cases,  and  con- 
cluded that  the  capacity  of  the  substance  for  caloric  had  not  been 
reduced  when  the  body  was  broken  up  into  smaller  portions. 

He  was  not  aware  that  there  was  a  distinct  difference  between  the 
physical  states  of  the  two  specimens  of  the  substance  which  might 
have  caused  his  experiment  to  indicate  a  wrong  result.  For  the 
filings  might  have  been  so  strained  as  to  contain  a  considerable 
amount  of  latent  heat  which  would  only  appear  on  their  complete 
recovery  from  the  state  of  strain.  His  conclusion  was  nevertheless 
correct,  for  it  is  entirely  supported  by  experiments  based  upon 
accurate  principles;  and  his  observations  therefore  prove  that  the 
caloric  hypothesis  is  incorrect. 

But  Kumford  did  not  stop  at  this  stage.  He  observed  that  the 
quantity  of  heat  which  was  developed  in  the  process  of  boring  was 
independent  of  the  amount  of  metal  which  was  abraded — that  a 
blunt  borer  and  a  sharp  borer,  though  producing  very  different 
amounts  of  filings,  caused  the  same  development  of  heat  if  the  same 
amount  of  work  were  spent  in  driving  them.  And,  further,  there 
seemed  to  be  practically  no  limit  to  the  amount  of  heat  which 
might  be  produced.  He  therefore  reasoned  that  heat  could  not  be  a 
material  substance,  and  stated  that  he  could  scarcely  conceive  of 
anything  but  motion  'which  could  be  excited  and  communicated  in 
the  manner  observed. 

Almost  at  the  same  time  (in  1799)  Davy  was  experimenting  in 
precisely  the  same  direction.  He  showed  that  two  pieces  of  ice 


THE    NATURE    OF    HEAT.  299 

might  be  melted  simply  by  rubbing  them  together.     Now  heat  is 
required  to  produce  this  change  of  state,  and  so,  on  the  caloric  hypo-  ( 
thesis,  the  capacity  of  water  for  heat  must  be  less  than  the  capacity 
of  ice  f&r  heat.     But  it  is  well  known  that  the  exact  reverse  is 
true. 

The  necessary  heat  might  have  been  furnished  by  surrounding 
bodies,  and  therefore  Davy  enclosed  the  two  pieces  of  ice  by  other 
portions  of  ice  and  placed  them  in  the  exhausted  receiver  of  an  air- 
pump.  Under  these  conditions  heat  could  only  reach  them  by  first 
melting  the  surrounding  ice. 

Davy  was  entitled  to  conclude,  from  the  result  of  his  experiments, 
that  heat  was  not  a  form  of  matter,  but,  at  the  time,  he  merely 
said,  'Friction,  consequently,  does  not  diminish  the  capacities  of 
bodies  for  heat.' 

In  1812,  when  again  discussing  the  point,  he  spoke  of  heat  as  '  a 
peculiar  motion,  probably  a  vibration,  of  the  corpuscles  of  bodies 
tending  to  separate  them,'  and  said  that  '  The  immediate  cause  of 
the  phenomenon  of  heat,  then,  is  motion,  and  the  laws  of  its  com- 
munication are  precisely  the  same  as  the  laws  of  communication  of 
motion.' 

The  second  statement  is  rigidly  correct ;  the  first — that  heat  is 
motion — is  only  true  if  properly  interpreted.  Heat,  since  it  is  not 
matter,  must  be  energy  ;  and  so  the  true  meaning  of  Davy's  state- 
ment is  that  heat  consists  in  the  energy  of  motion  of  the  particles  of 
a  material  body.  But  the  word  *  energy  '  was  not  introduced  into 
science  at  that  time. 

Davy  illustrated  this  heat-motion,  which  he  termed  '  repulsive 
motion,'  by  the  analogy  of  the  orbital  motion  of  the  planets.  If  the 
speed  of  motion  of  any  planet  were  increased,  the  orbit  would 
become  larger  just  as  if  a  repulsive  force  had  acted. 

In  Davy's  statement  we  have  therefore  the  complete  foundation 
of  the  whole  kinetic  theory  of  gases  (Chap.  XIII.),  and,  more 
generally,  of  the  modern  dynamical  theory  of  heat. 

Since  heat  is  a  form  of  energy  we  might  infer  the  possibility  of  its 
existence  in  a  potential  form,  and  the  use  of  the  common  term  latent 
heat  bears  out  the  inference. 

In  working  out  a  dynamical  theory  of  heat  Rankine  advanced  a  hypo- 
thesis of  '  molecular  vortices.'  He  supposed  that  the  motions  which 
constitute  heat  in  bodies  are  vortical  motions  in  atmospheres  around 
nuclei,  and  that  radiation  consists  in  the  propagation  of  vibratory 
motions  of  the  nuclei  under  their  mutual  forces.  The  energy  of  the 
vortices  is  the  amount  of  heat  which  bodies  possess  ;  and  the  absolute 
temperature  of  any  body  is  the  quotient  of  this  energy  by  a  definite 


330  A   MANUAL    OF   PHYSICS. 

constant  for  each  substance.  The  elastic  pressure,  according  to 
dynamical  laws,  must  be  directly  proportional  to  the  vortical  energy, 
and  must  be  inversely  proportional  to  the  volume  which  the  vortices 
occupy,  except  in  so  far  as  the  mutual  nuclear  forces  which  exist  in 
all  non-perfect  gases  modify  this  result.  Latent  heat  is  the  equiva- 
lent of  work  done  in  varying  the  dimensions  of  the  vortical  orbits 
when  the  volumes  and  shapes  of  the  spaces  which  they  occupy  are 
altered.  Specific  heat  is  the  equivalent  of  work  spent  in  varying  the 
vortical  energy. 


CHAPTEK  XXI. 

RADIATION   AND   ABSORPTION   OF   HEAT. 

255.  Prevost's  Theory  of  Exchanges. — The  fact  that  the  laws  of 
communication  of  heat  are  precisely  those  of  the  communication  of 
motion  leads  to  the  conclusion  that  motion  of  the  particles  of  a  hot 
body  is  not  confined  to  the  surface  alone ;  and  this  conclusion  is 
confirmed  by  the  greater  radiation  which  takes  place  from  a  thick 
plate,  than  from  a  thin  plate,  of  a  transparent  substance,  when  both 
plates  are  at  the  same  temperature.  It  indicates  also  that  the 
radiation  from  a  hot  body  is  dependent  upon  the  state  of  that  body 
alone,  and  is  not  influenced  by  the  presence  of  any  other  body, 
except  in  so  far  as  it  may  cause  an  alteration  in  the  thermal  state  of 
the  former. 

This  is  the  essence  of  the  Theory  of  Exchanges  which  was 
advanced  by  Prevost  of  Geneva,  under  the  title  of  a  theory  of 
'  movable  equilibrium  of  temperature.' 

According  to  Prevost,  two  bodies,  which  are  of  different  tempera- 
tures, and  are  placed  in  an  enclosure  which  is  impervious  to  heat, 
will  both  radiate  heat.  The  hotter  body  will  radiate  at  a  greater 
rate  than  the  other,  until,  by  absorption,  the  temperatures  of 
the  two  are  equalised.  After  this,  each  will  still  radiate,  but  at 
precisely  the  same  rate  ;  so  that  the  heat  which  one  loses  by 
radiation  is  balanced  by  that  which  it  gains  by  absorption.  This  is 
the  condition  which  Prevost  termed  a  condition  of  movable 
equilibrium  (or,  as  we  would  now  call  it,  of  kinetic  equilibrium)  of 
temperature. 

256.  Stewart's  and  Kirchhoff"  s  Extension  of  Prevost' s  Theory. — 
As  in  Chap.  XVII.,  we  define  the  absorptive  power  of  a  body,  undei 
given  conditions,  for  any  definite  radiation,  as  the  fraction  of  the 
whole  incident  radiation  of  that  kind  which  it  absorbs  ;  and  we  also 
define  the  emissivity  of  a  body,  at  a  given  temperature,  for  any 
given  radiation,  as  the  ratio  of  the  quantity  of  that  radiation  which 


302  A   MANUAL   OF    PHYSICS. 

it  emits  to  the  quantity  of  it  which  is  emitted  by  a  black  body  under 
the  same  conditions. 

And  Stewart's  proof,  as  given  in  Chap.  XVII.,  leads  to  the  result 
that  the  emissivity  and  the  absorptive  power  of  a  body,  at  a  given 
temperature,  for  any  radiation,  are  equal. 

It  is  needless  to  enter  into  any  discussion  of  special  cases  further 
than  those  which  have  already  been  given  (§  203).  Suffice  it  to  say 
that,  whatever  be  the  quality  and  the  quantity  of  the  radiation 
emitted  by  any  body  under  given  conditions,  absorption  must  exactly 
balance  emission,  in  respect  both  of  quality  and  quantity,  if  the 
given  conditions  are  to  be  maintained.  Previously  to  Stewart's 
investigations,  it  was  known  from  the  experiments  of  Leslie,  De  la 
Provostaye,  and  Desains,  that  the  radiating  and  the  absorbing 
powers  of  any  one  body  were  proportional  to  each  other  ;  that  is  to 
say,  it  was  known  that  a  good  radiator  was  a  good  absorber,  and  that 
a  bad  radiator  was  deficient  in  absorbing  power. 

257.  Laws  of  Radiation  of  Heat  at  Constant  Temperature.  — 
Early  in  the  history  of  the  subject,  it  was  known  that  the  radiation 
from  a  body  at  a  given  temperature  depended  upon  the  nature  of  the 
surface  of  that  body.  (This  furnishes  an  additional  analogy  between 
light  and  radiant  heat.) 

Leslie  constructed  a  hollow  metal  cube,  one  side  of  which  was 
polished,  while  another  was  rough.  A  third  side  was  covered  with 
lampblack,  and  the  fourth  was  coated  with  white  enamel.  Although 
the  surfaces  of  the  two  latter  sides  were  so  very  different,  Leslie 
found  that  both  radiated  about  equally  well  when  the  cube  was 
filled  with  hot  water.  The  radiation  from  the  bright  metallic  sur- 
face was  much  smaller  than  that  from  any  of  the  others,  and  the 
radiation  from  the  rough  metallic  surface  was  considerably  less  than 
that  from  the  enamelled  and  the  blackened  surfaces. 

It  has  already  been  proved  (§  205)  that  the  amount  of  any  given 
radiation,  which  is  emitted  from  a  sufficient  thickness  of  a  substance 
of  given  radiating  power,  is  equal  to  that  which  is  emitted  from  a 
black  body  at  the  same  temperature.  It  was  shown  that  the 
amount  which  is  transmitted  through  a  plate  of  the  substance,  n 
units  in  thickness,  is  K(l—  p)",  where  E  is  the  total  incident  radia- 
tion and  p  is  the  absorption  co-efficient.  Hence  the  amount  which 
is  stopped  by  the  plate  is 


and  therefore,  by  definition,  the  absorptive   power  —  and,   conse- 
quently, the  emissivity  —  is 

l-(l-p)'. 


RADIATION    AND    ABSORPTION    OF    HEAT.  303 

As  the  temperature  of  a  body  rises,  radiations  of  shorter  and 
shorter  wave-length  are  emitted,  and  the  energy  in  each  pre- 
viously existing  kind  is  increased. 

258.  Heat  Spectra. — If  the  radiation  from  a  luminous  body  be 
passed  through  a  slit  and  a  prism  in  the  usual  manner,  a  spectrum 
is  obtained  which  enables  us  at  once  to  discover  the  nature  of  the 
radiation.  And  we  may  measure  the  amount  of  energy  in  any 
given  portion  of  the  spectrum  by  allowing  that  radiation  to  fall  upon 
a  medium  which  entirely  absorbs  it,  and  is  consequently  raised  in 
temperature  to  a  measurable  extent.  But,  in  any  such  experiment, 
it  is  necessary  first  to  make  certain  that  absorption  does  not  take 
place,  to  any  appreciable  extent,  in  the  substance  of  the  prism. 

We  might  also  indirectly  analyse  the  radiation  emitted  by  a  given 
body  by  means  of  determinations  of  the  absorption  which  the  body 
exercises  upon  radiations  of  different  wave-lengths;  but  such 
measurements  are  of  little  value  unless  the  substance  used  is  of 
definite  chemical  composition  and  physical  structure.  Some  gases 
(e.g.,  olcfiant  gas)  exert  powerful  absorption  on  the  heat-rays ; 
others -exhibit  very  little.  The  absorption  produced  by  water  vapour 
seems  to  be  largely  due  to  the  dust  nuclei  (§  277). 

These  remarks  apply  to  the  invisible  portions  of  the  spectrum 
also,  whether  these  consist  of  rays  of  higher  refrangibility,  or  of 
rays  of  lower  refrangibility,  than  those  which  form  the  visible 
parts. 

It  is  found  that  the  invisible  portions  of  the  spectrum  possess 
characteristics  which  are  precisely  analogous  to  those  which  appear 
in  the  visible  parts. 

We  cannot,  by  means  of  the  thermopile  (§  328),  or  of  the 
bolometer  (§  343),  which  are  the  two  most  suitable  instruments  for 
the  present  purpose,  determine  the  energy  of  the  radiation  of  one 
definite  wave-length.  All  that  can  be  done  is  to  measure  the 
energy  of  the  total  radiation  which  is  contained  between  rays  of 
known  wave-length,  for  the  face  of  the  thermopile  and  the  metal 
strip  of  the  bolometer  necessarily  possess  considerable  breadth. 

In  the  case  of  a  refraction  spectrum,  besides  the  difficulty  regard- 
ing absorption  by  the  substance  of  the  prism,  there  is  the  difficulty 
of  the  crowding  together  of  the  rays  towards  the  less  refrangible 
end  of  the  spectrum  according  to  an  unknown  law. 

If  a  diffraction  spectrum  be  used,  absorption  may  take  place  in 
the  substance  of  the  grating  if  it  be  made  of  glass  ;  and,  if  it  be  a 
metallic  grating,  great  uncertainty  exists  with  regard  to  the  nature 
of  its  action  upon  the  invisible  rays.  And,  although  the  dispersion 
be  (§  233)  practically  proportional  to  the  wave-length,  so  that  equal 


304  A    MANUAL    OF    PHYSICS. 

breadths  of  the  spectrum  correspond  to  equal  differences  of  wave- 
length, we  must  remember  that  the  measuring  instruments  ought  to 
determine  the  energy  contained  in  portions  of  the  spectrum  which 
are  bounded  by  rays  the  wave-lengths  of  which  are  in  a  constant 
ratio.  Further,  since  we  are  dealing  with  radiations  of  all  wave- 
lengths, we  see  that,  at  any  one  part  of  a  diffraction  spectrum,  an 
infinite  number  of  different  radiations  are  superposed,  because  of 
the  existence  of  an  infinite  number  of  spectra  of  different  orders. 

The  difficulties  of  the  problem  have  been  largely  overcome  by 
Langley,  to  whom  is  chiefly  due  our  recent  great  increase  of  know- 
ledge regarding  radiations  of  large  wave-length.  He  has  detected, 
by  means  of  the  bolometer,  traces  of  heat  in  portions  of  the  solar 
spectrum  corresponding  to  wave-lengths  about  twenty-four  times 
greater  than  those  of  the  least  refrangible  part  of  the  visible 
spectrum. 

He  has  shown  that  the  wave-length  at  which  the  maximum  of 
energy  in  the  spectrum  exists  diminishes  as  the  temperature  is 
raised.  This  result  has  been  deduced  from  theoretical  considera- 
tions by  Michelson  ;  and  the  numerical  deductions  from  the  theory 
accord  very  well  with  Langley's  observations.  The  energy  at  a 
given  part  of  the  spectrum  dies  away  in  amount  very  rapidly 
when  we  pass  from  the  maximum  in  the  direction  of  decreasing 
wave-length;  it  dies  away  much  more  slowly  as  we  pass  in  the 
opposite  direction  along  the  spectrum. 

Cauchy's  formula  (§  209)  connecting  the  refractive  index  of  a  sub- 
stance for  a  given  radiation  with  the  wave-length  of  that  radiation 
does  not  agree  with  Langley's  observations  on  the  heat-rays.  Briot's 
formula  agrees  better,  but  it  ultimately  differs  from  them  in  the 
opposite  direction  to  that  in  which  Cauchy's  differs  from  them. 

259.  Radiation  at  Different  Temperatures. — Hitherto  we  have 
assumed  that  the  radiating  bodies  with  which  we  have  dealt  have 
been  kept  at  constant  temperature  so  that  their  radiating  powers 
remained  unaltered.  We  must  now  consider  the  relation  between 
emissivity  and  temperature. 

Let  us  suppose  that  the  hot  body  is  placed  inside  an  enclosure 
which  is  kept  at  a  constant  temperature  t.  Let  t-\-Q  be  the 
temperature  of  the  hot  body,  and  let  no  heat  pass  from  it  except  by 
radiation. 

If  f(t)  represent  the  rate  of  loss  of  heat  from  the  hot  body  at  tem- 
perature t,  the  rate  of  loss  of  heat  under  the  assumed  conditions 
will  be 

/(*  +  0)-/(0; 

and,  since  we  have  no  adequate  knowledge  of  the  mechanism  of 


RADIATION   AND   ABSORPTION   OF   HEAT.  305 

emission,  the  form  of  this  expression  must  be  determined  from 
experiment. 

According  to  Newton,  the  rate  of  loss  of  heat  is  proportional  to 
the  excess  of  the  temperature  of  the  hot  body  over  that  of  its  sur- 
roundings ;  or,  in  symbols, 


But,  since  we  must  regard  the  rate  of  emission  as  independent  of  the 
surrounding  bodies,  this  is  equivalent  to 

f(t)  =  at  +  b, 

where  a  and  &  are  constants,  and  t  is  any  temperature. 

The  above  law  is  not  even  roughly  accurate  unless  the  differences 
of  temperature  are  small,  and  it  becomes  less  and  less  applicable 
the  greater  the  differences  are. 

Dulong  and  Petit  made  an  elaborate  series  of  experiments  with 
the  view  of  discovering  a  more  correct  law.  They  found  that,  when 
the  temperature  difference  was  kept  constant,  the  rate  of  loss  in- 
creased in  geometrical  progression  as  the  temperature  of  the  sur- 
roundings of  the  body  increased  in  arithmetical  progression  ;  and 
the  ratio  of  the  geometrical  series  is  independent  of  the  tempera- 
ture excess  —  so  long,  at  least,  as  the  excess  is  not  greater  than 
200°  C. 

When  the  temperature  excess  vanishes,  the  loss  of  heat  is  zero. 
Dulong  and  Petit  therefore  expressed  the  law  in  the  form 


which  agrees  very  closely  with  the  result  of  their  observations  when 
t  varies  from  0°  up  to  80°  C.,  and  9  does  not  exceed  200°  C.  Since 
this  formula  may  be  written  in  the  form 

/(*  +  e)  -/(*)  =  »  ««+*-<&  < 

we  see  that  the  absolute  rate  of  radiation*  independently  of  the 
surroundings,  is 

f(t)  =  aa'+  b. 

The  constant  a  depends  on  the  nature  of  the  radiating  surf  ace  t 
but  the  constant  a  is  practically  an  absolute  constant,  and  is  equal 
to  1-0077.  The  law  of  Dulong  and  Petit,  therefore,  asserts  that, 
when  the  temperature  excess  is  constant,  the  rate  of  loss  of  heat  is 
proportional  to  (1*0077)*,  where  t  is  the  absolute  temperature  of  the 

20 


306  A   MANUAL    OF    PHYSICS. 

bodies  to  which  the  heat  is  radiated.     It  asserts  also  that  the  rate  of 
loss  is  proportional  to  (1*0077)  0  —  1  when  t  is  constant. 
The  equation 


where  r  represents  rate  of  loss,  might  not  have  been  borne  out  by 
experiment.  The  fact  that  it  is  so  borne  out  furnishes,  as  Balfour 
Stewart  pointed  out,  an  independent  proof  of  the  truth  of  Prevost's 
Theory  of  Exchanges,  of  which  it  is  a  necessary  consequence. 

We    may  write  the  expression  a®—  1  in  the  form  (1  -{-  _£>)#  —  1, 
which,  by  expansion,  becomes 


But,  by  Newton's  law  of  cooling,  the  rate  of  loss  should  be  propor- 
tional to  9  ;  and  hence,  substituting  the  values  9  —  11,  p  =  0-0077, 
we  see  that  Newton's  value  is  fully  4  per  cent,  too  small  when  the 
temperature  excess  is  11°. 

More  recent  experiments  by  De  la  Provostaye  and  Desains  verified 
the  accuracy  of  the  law  of  Dulong  and  Petit  within  the  limits 
already  assigned. 

According  to  Hopkins  the  radiation  per  square  foot  per  minute, 
from  glass  at  100°  C.  to  an  enclosure  at  0°  C.,  is  0-176  heat  units, 
the  unit  being  the  amount  of  heat  which  is  required  to  raise  the  tem- 
perature of  one  pound  of  water  from  0°  C.  to  1°  C.  Under  the  same 
conditions  the  radiation  from  unpolished  limestone  is  0'236  units  ; 
and  that  from  polished  limestone  cut  from  the  same  block  is  0'168 
units. 

Dulong  and  Petit's  law  seems  to  be  applicable  only  to  the  total 
radiation  from  a  body,  and  not  to  each  definite  radiation  of  which 
the  whole  is  composed.  The  rate  of  emission  of  particular  radia- 
tions from  a  black  body  seems  to  increase  rapidly  at  first,  and  then 
more  slowly,  as  the  temperature  of  the  black  body  is  raised. 

260.  Solar  Radiation.  —  Pouillet  was  the  first  to  make  fairly 
accurate  measurements  of  the  amount  of  radiation  which  is  received 
by  the  earth  from  the  sun  in  a  given  time.  For  this  purpose  he 
invented  the  Pyrliclioinctcr. 

This  instrument  consists  of  a  flat  cylindrical  metallic  vessel,  the 
surface  of  which)  with  the  exception  of  one  end  which  is  covered 
with  lamp-black,  is  highly  polished.  The  bulb  of  a  thermometer  is 


RADIATION   AND   ABSORPTION    OF   HEAT.  307 

inserted  in  the  cylinder,  and  its  stem  lies  along  the  axis.  The 
cylinder  is  filled  with  water  or  mercury,  and  its  blackened  face  is 
directed  towards  the  sun.  In  order  that  this  may  be  done  with 
accuracy,  a  metal  disc,  the  diameter  of  which  is  exactly  equal  to 
that  of  the  cylinder,  is  fixed  on  the  end  of  the  axis  of  the  instru- 
ment remote  from  the  cylinder.  The  shadows  of  the  disc  and  the 
cylinder  will  not  coincide  unless  the  face  of  the  latter  be  accurately 
turned  towards  the  sun.  The  area  of  the  blackened  face,  and  the 
amount  of  heat  necessary  to  raise  the  temperature  of  the  cylinder 
and  its  contents  to  a  given  extent,  are  exactly  determined. 

If  the  temperature  of  the  instrument  is  the  same  as  that  of  the  air, 
and  if  its  blackened  face,  carefully  shaded  from  the  sun,  be  turned 
towards  the  sky,  radiation  will  take  place,  and  the  temperature  will 
fall  9°  (say)  in  t  units  of  time. 

Now  let  the  apparatus  be  turned  towards  the  sun  for  t  units  of 
time  ;  after  which  let  it  be  turned  towards  the  sky  as  before,  during 
an  equal  time,  and  let  9f  be  the  fall  of  temperature. 

It  is  concluded  that  the  deficiency  introduced  into  the  total  rise 
of  temperature  which  took  place  when  the  instrument  was  exposed 
to  the  sun,  is 


for,  during  this  exposure,  the  cylinder  was  steadily  rising  in  tem- 
perature because  of  the  heat  which  it  absorbed  from  the  sun,  and 
was  at  the  same  time  steadily  falling  in  temperature  because  of  its 
own  radiation. 

Hence  the  full  rise  of  temperature,  had  no  radiation  taken  place 
from  the  cylinder,  would  have  been 


-. 

where  6  is  the  rise  of  temperature  which  was  actually  observed.    And 
so,   from  the   known   constants   of   the  instrument,   the   quantity 
of  heat  which  was  received,  in  a  given  time,  by  unit  area  at  the 
earth's  surface  could  be  calculated. 
Pouillet  gave  the  expression 


for  the  rate  of  rise  of  temperature  under  the  above,  conditions  at 
different  times  of  the  day.  The  quantity  I  is  the  thickness  of  the 
earth's  atmosphere  which  was  traversed  by  the  sun's  rays.  The 
constant  6  varies  from  day  to  day,  but  the  constant  a  does  not  so 
vary.  Pouillet  concluded  that  the  expression  would  be  applicable 

20—2 


dOH  A    MANUAL    OF    PHYSICS. 

even  if  there  were  no  atmosphere,  in  which  case  the  constant  a 
would  represent  the  rate  of  rise  of  temperature  ;  and  he  calculated 
that,  in  the  event  of  no  atmospheric  absorption,  the  quantity  of  heat 
falling  on  a  square  centimetre  of  the  earth's  surface,  in  one  minute, 
would  raise  the  temperature  of  1*76  grammes  of  water  by  1°  C. 

From  the  average  values  of  the  constant  e,  Pouillet  inferred  that 
about  one -half  of  the  total  incident  solar  radiation  is  absorbed  in  the 
earth's  atmosphere.  Sir  W.  Thomson  concludes,  from  the  above 
data,  and  those  independently  given  by  Herschel,  that  the  rate  of 
radiation  from  the  sun's  surface  is  about  7,000  horse-power  per 
square  foot,  or  thirty  times  the  amount  which  is  radiated,  per  square 
foot,  from  the  furnace  of  a  locomotive. 

The  Actinometer  is  another  instrument  used  to  determine  the 
magnitude  of  solar  radiation.  It  consists  essentially  of  a  metal 
enclosure  (blackened  internally  and  kept  at  constant  temperature) 
at  the  centre  of  which  the  bulb  of  a  thermometer  is  placed.  By 
means  of  an  opening  in  the  enclosure,  the  sun's  rays  are  allowed  to 
fall  upon  the  bulb  for  a  given  time,  after  which  the  bulb  radiates  to 
the  enclosure.  Various  experimenters  have  used  this  apparatus  in 
one  or  other  of  its  forms.  By  its  means  Violle  has  found  that  the 
radiation  from  the  sun,  which  would  fall  per  minute  on  a  square 
centimetre  of  the  earth's  surface  did  no  absorption  occur,  would 
raise  the  temperature  of  2-54  grammes  of  water  by  1°  C. 

Langley's  more  recent  measurements  also  indicate  that  Pouillet's 
estimate  is  too  low.  He  finds  that  the  quantity  of  unabsorbed  heat 
falling  on  a  square  centimetre  would  raise  the  temperature  of 
1-81  grammes  of  water  by  1°  C.  Did  no  absorption  take  pl&ce,  this 
would  become  2'8  grammes,  and  the  amount  of  heat  radiated  from 
the  sun  per  square  foot  of  its  surface  would  be  fifty  times  greater 
than  that  radiated  from  a  square  foot  of  the  surface  of  a  loco- 
motive. 

From  Langley's  data  we  can  calculate  that  the  yearly  radiation 
from  the  sun  to  the  earth  would,  if  spread  uniformly  over  the  surface, 
melt  a  uniform  crust  of  ice  fully  150  feet  in  thickness. 

261.  Radiation  from  Moving  Bodies. — The  following  considera- 
tions regarding  the  motion  of  radiating  bodies,  first  advanced  by 
Balfour  Stewart,  are  specially  worthy  of  notice — apart  from  their 
intrinsic  value — as  an  example  of  the  useful  employment  of  the 
principle  of  Conservation  of  Energy. 

Let  us  suppose  that  ultimate  equality  of  temperature  has  been 
arrived  at  inside  the  enclosure,  impervious  to  heat,  which  was  pos- 
tulated in  §  203  ;  and  let  us  further  assume  that  one  of  the  bodies  in 
that  enclosure  suddenly  commences  to  move  about  with  a  speed 


RADIATION    AND    ABSORPTION    OF    HEAT.  309 

which  is  comparable  with  that  of  light.  A  direct  application  of 
Doppler's  principle  shows  us  that  any  other  body  in  the  enclosure, 
towards  which  the  motion  may  be  directed,  will  receive  energy  from 
the  moving  body  at  a  greater  rate  than  will  one  which  is  so  situated 
that  the  motion  is  directed  from  it. 

It  follows  that  relative  motion  of  radiating  bodies  is  inconsistent 
with  ultimate  equality  of  temperature.  But  persistent  inequality  of 
temperature  would  imply  a  perpetual  source  of  energy;  and  we 
therefore  conclude  that  the  relative  motion  of  radiating  bodies  must 
gradually  cease. 


CHAPTEK  XXII. 

EFFECTS  OF   THE  ABSORPTION  OF   HEAT:    DILATATION   AND   ITS 
PRACTICAL    APPLICATIONS. 

262.  Temperature. — Increase  of  temperature  usually  accompanies 
the  application  of  heat  to  any  substance.  The  fundamental  distinc- 
tion between  heat  and  temperature  is  very  clearly  brought  out  by  a 
consideration  of  the  meaning  of  the  words  hot  and  cold  as  applied 
to  different  material  substances.  The  bodies  which  are  said  to  be 
hot  or  cold  may  really  be  at  the  same  temperature.  Thus  a  mass  of 
iron  and  a  mass  of  wood,  though  their  temperatures  be  equal,  feel 
very  differently  to  the  touch.  If  the  temperature  of  the  hand  be 
higher  than  that  of  the  two  masses,  the  former  feels  cold,  and  the 
latter  feels  warm ;  while,  if  the  temperature  of  the  hand  be  lower 
than  that  of  the  bodies,  these  conditions  are  reversed.  The  reason 
s  that  the  physical  properties  of  the  substances  are  such  that,  of  the 
two,  iron  is  the  one  which  most  rapidly  abstracts  heat  from,  or 
supplies  heat  to,  the  hand. 

It  is  sufficient  for  our  present  purpose  that  we  regard  temperature 
as  that  condition  which  determines  the  flow  of  heat  from  one  body 
to  another.  (See  §§  254,  150.) 

If  two  bodies,  which  differ  in  temperature,  be  placed  in  contact 
with  each  other  (or  in  thermal  communication  of  any  sort),  heat 
passes  from  the  body  which  is  at  the  higher  temperature  to  the  body 
at  the  lower  temperature.  And  two  bodies,  between  which,  on  the 
whole,  there  is  no  transference  of  heat  when  thermal  communica- 
tion is  established,  are  said  to  have  equal  temperatures. 

It  is  an  experimental  fact  that  any  two  bodies,  which  have  each 
the  same  temperature  as  a  third  body,  are  themselves  at  equal 
temperatures. 

Various  methods,  which  will  be  described  later,  are  used  for  the 
determination  of  the  difference  of  temperature  which  subsists 
between  two  bodies,  or  between  a  body  in  one  given  physical  state 
and  the  same  body  when  in  another  state. 


DILATATION   AND   ITS   PRACTICAL   APPLICATIONS.  311 

We  may  premise,  for  present  purposes,  that  the  temperatures  of 
ice-cold  water,  and  of  boiling  water,  respectively,  are  called  0  degrees 
and  100  degrees  ;  that  a  given  number  of  degrees,  at  any  part  of  our 
scale,  corresponds  to  a  constant  interval  of  temperature  ;  and  that 
we  call  the  degrees  Centigrade  degrees  —  the  letter  C  being  used  to 
discriminate  them  from  the  degrees  of  other  scales. 

263.  Dilatation  of  Solids.  —  One  of  the  most  obvious  effects  of  the 
application  of  heat  to  a  substance  is  expansion.  In  a  homogeneous 
isotropic  solid,  the  expansion  is  equal  in  all  directions.  On  the 
other  hand,  in  a  non-isotropic  solid,  the  expansion  is  different  in 
different  directions  ;  but,  in  this  case,  three  rectangular  directions 
(called  the  principal  axes,  cf.  §  245),  can  always  be  found  such  that 
in  one  the  expansion  is  a  maximum  ;  in  another,  it  is  a  minimum  ; 
and,  in  the  third,  its  value  is  intermediate  between  those  along  the 
other  two  —  being,  in  fact,  a  maximum-minimum.  When  the 
expansions,  along  these  directions,  which  accompany  a  given  rise  of 
temperature,  are  known,  the  expansion  in  any  other  direction  can 
be  found. 

We  have,  first  of  all  then,  to  consider  the  laws  of  linear  dilatation 
of  a  solid.  [The  requisite  measurements  are  readily  made  by  direct 
micrometric  methods,  which  give  accurate  determinations  of  the 
lengths  of  a  bar  at  different  known  temperatures.] 

1.  It  is  found  that  the  increase  of  length  of  a  given  bar  is  pro- 
portional to  the  length  of  the  lor  at  the  initial  temperature. 

2.  The  alteration  of  length  is  proportional  to   the  increase  of 
temperature. 

These  laws  are  symbolically  represented  by  the  equation 


where  lt  and  lu  are  respectively  the  lengths  of  the  bar  at  the  higher 
and  lower  of  the  two  temperatures  of  which  t  is  the  difference,  and 
k  is  a  constant  called  the  co-efficient  of  linear  dilatation.  This 
constant  is  obviously  the  increase  of  length,  per  unit  rise  of  tempera- 
ture, of  a  bar  the  original  length  of  which  is  unity. 
From  the  three  equations 


where  the  letters  Z,  6,  and  d  denote  the  length,  breadth,  and  thick- 
ness of  a  rectangular  parallelepiped  of  the  substance,  and  &„  &2,  Tc 
are  respectively  the  co-efficients  of  dilatation  measured  in  the  direc- 
tion of  the  length,  breadth,  and  thickness,  we  can  obtain  an  expression 


312  A   MANUAL   OF   PHYSICS. 

for  the  cubical  dilatation  of  the  solid.     Let  us  suppose  that  these 
equations  refer  to  the  principal  axes  ;  then 


or 


where  the  Vs  represent  volumes. 

In  all  cases  in  nature,  the  quantities  lsv  &2,  &3  are  so  small  that 
their  squares  and  products  may  be  neglected  ;  hence,  to  a  sufficient 
degree  of  approximation, 


if  K  represents  the  co-efficient  of  cubical  dilatation.     This  gives 

K  =  M-*a+fcs; 

and  so  the  co-efficient  of  cubical  dilatation  (which  is  the  fractional 
increase  in  bulk  of  unit  volume  per  unit  rise  of  temperature)  is 
equal  to  the  sum  of  the  three  principal  co-  efficients  of  dilatation, 
When  the  substance  is  isotropic,  this  latter  equation  becomes 

K  =  3&; 

which  asserts  that  the  co-efficient  of  cubical  dilatation  of  a  homo- 
geneous isotropic  solid  is  three  times  the  co-efficient  of  linear  dilata- 
tion. 

If  the  edges  of  the  rectangular  parallelepiped  be  not  in  the  direc- 
tions of  the  three  principal  axes,  the  effect  of  the  application  of 
heat  will  be  to  change  the  inclination  of  the  faces,  so  that  the 
parallelepiped  will  cease  to  be  rectangular.  When  the  three  edges 
are  all  originally  equal,  and  the  diagonals  of  one  of  the  square 
faces  of  the  cube  are  parallel  to  two  of  the  principal  axes,  the 
application  of  heat  changes  the  square  face  into  a  rhombus  such 
that  the  tangent  of  half  its  obtuse  angle  is  equal  to  1  +  (7ci  —  Jc2)t, 
where  t  is  the  increase  of  temperature  and  the  original  length  of  the 
diagonals  is  assumed  to  be  unity.  This  affords  a  ready  means  of 
determining  the  co-efficient  of  expansion  along  one  of  the  two 
principal  axes,  provided  that  we  know  its  value  along  the  other  ;  for 
the  change  of  angle  can  be  measured  with  extreme  accuracy  by 
optical  methods. 

Fizeau  introduced  a  specially  delicate  method  of  measuring  the 
change  of  length  of  a  rod.  Newton's  rings  (§  221)  are  produced  by 
the  interference  of  light  reflected  at  the  surface  of  a  film  of  air 
contained  between  two  plates  of  glass,  of  which  one  at  least  is 
slightly  curved  ;  and  the  colour  of  any  particular  ring  depends  on 
the  thickness  of  the  film,  which  can  be  calculated  when  the  radius 


DILATATION   AND    ITS    PRACTICAL    APPLICATIONS,  313 

of  the  ring  is  known.  If  one  of  the  two  plates  be  fixed,  while  the 
other  is  attached  to  the  expanding  rod,  the  slightest  change  of 
length  of  the  rod  causes  a  diminution  of  the  thickness  of  the  film 
of  air  which  can  be  easily  calculated  by  means  of  the  optical 
changes  which  are  simultaneously  produced. 

He  gives  the  following  empirical  formula  connecting  the  co- 
efficient k  with  the  temperature  t  expressed  in  Centigrade  degrees  : 

fc  =  a  +a'(t-  40). 

The  constants  were  determined  by  observations  at  three  tempera- 
tures, viz.,  10°,  45°,  and  70°. 

The  cubical  dilatation  of  a  solid  is  usually  determined  directly  by 
heating  it  in  a  liquid,  of  known  expansibility,  which  is  contained  in 
a  vessel  made  of  a  substance  whose  co-efficient  of  linear  (and,  there- 
fore, also  of  cubical)  dilatation  has  been  found.  Let  V0  be  the 
volume  of  the  vessel  at  0°  C.,  and  let  KX  be  its  co-efficient  of  dilata- 
tion. The  volume  at  t°  C.  is 


Similarly,  the  volume  at  t°  C.  of  the  liquid  which  filled  the  vessel  at 
0°  C.  is 


If  we  now  place  a  solid,  the  volume  of  which  at  0°  C.  is  voj  and  at 

t°  C.  is 


the  volume  of  liquid  which  overflows  from  the  vessel  at  tempera- 
ture t  is 

V,(1+IM)  - 


or  VXKj 

from  which  expression  the  value  of  K3  may  be  found. 

It  is  needless  to  enumerate  the  various  practical  applications  of 
the  dilatation  of  solid  bodies  when  their  temperature  is  raised,  or  of 
their  contraction  as  the  temperature  falls.  The  well-known 
processes  of  shrinking  the  tires  on  wheels,  and  of  drawing  together 
the  walls  of  a  building  when  these  have  bulged  outwards,  will 
sufficiently  serve  as  an  indication  of  their  nature. 

One  particular  application  —  to  the  construction  of  a  compensation- 
pendulum  or  balance-wheel  of  a  watch  —  merits  special  notice. 

The  ordinary  compensation-pendulum  is  constructed  upon  the 
principle  that  the  difference  between  the  lengths  of  two  rods,  of 
different  expansibilities,  will  remain  constant,  however  the  tempera- 
ture may  be  altered  within  allowable  limits,  provided  that  the  lengths 


314  A   MANUAL    OF    PHYSICS. 

of  the  rods  be  made  in  inverse  proportion  to  the  co-efficients  of 
dilatation  of  the  substances  of  which  they  are  composed.  This 
obviously  affords  a  means  of  keeping  constant  the  distance  between 
the  bob  of  a  pendulum  and  its  point  of  support. 

If  two  equal  bars,  of  different  expansibilities,  be  soldered  together 
throughout  their  length,  a  rise  of  temperature  will  cause  the  com- 
pound bar  to  bend  in  such  a  way  that  the  less  expansible  bar  is  on 
the  concave  side ;  for  this  is  the  only  way  in  which  the  tendency  to- 
wards unequal  expansions  can  be  satisfied.  This  fact  is  made  use 
of  in  the  construction  of  compensated  balance  wheels.  When  the 
temperature  of  an  uncompensated  wheel  increases,  the  expansion  of 
spokes  carries  the  rim  of  the  wheel  further  out  from  the  centre,  and 
the  consequent  increase  of  moment  of  inertia  produces  an  increase 
in  the  period  of  oscillation.  Hence  a  watch,  or  chronometer,  fitted 
with  such  a  wheel  will  go  too  slow  in  warm  weather  and  too  fast  in 
cold  weather.  But  if  the  rim  of  the  wheel  be  divided  into  a  number 
of  independent  parts,  each  part  being  carried  by  a  separate  spoke, 
and  if  the  rim  be  compound  as  above  described,  matters  may  be  so 
arranged  that  the  throwing-out  of  the  weight  from  the  centre  because 
of  the  expansion  of  the  spokes  is  counterbalanced  by  means  of  the 
inward  bending  of  the  segments  of  the  rim. 

It  is  worthy  of  note  that  one  or  two  of  the  principal  expansibilities 
of  a  solid  may  be  negative,  i.e.,  the  substance  may  contract  in  at 
least  one  direction  when  its  temperature  is  raised.  In  such  a  case 
a  series  of  directions  may  be  found  in  the  substance  such  that  a 
change  of  temperature  does  not  give  rise  to  any  alteration  of  length 
of  the  substance  in  any  of  these  directions.  [These  directions  may 
be  found  by  imagining  a  sphere  to  be  drawn  in  the  unheated  body 
and  finding  its  intersection  with  the  ellipsoidal  surface  into  which  it 
becomes  distorted  by  the  application  of  heat.  All  lines  drawn  in  the 
body  parallel  to  the  lines  joining  the  points  of  the  curves  of  inter- 
section to  the  centre  of  the  sphere  are  unchanged  in  length.] 
Hence  a  rod  cut  from  the  substance  in  any  such  direction  may  be 
used  as  the  rod  of  a  compensated  pendulum.  Brewster  pointed  out 
that  a  rod  of  marble  might  be  so  used. 

An  interesting  example  of  contraction  of  a  solid  when  its  tempera- 
ture is  raised  is  seen  in  the  case  of  india-rubber  under  tension.  The 
experiment  may  readily  be  performed  by  blowing  steam  through  a 
hollow  tube  of  the  substance,  which  is  fixed  at  its  upper  end,  and  is 
extended  by  means  of  a  weight  attached  to  the  lower  end. 

The  subjoined  table  contains  Fizeau's  determinations  (see  above) 
of  the  values  of  the  constants  in  his  formula  for  the  co-efficients  of 
linear  dilatation  of  a  few  well-known  substances.  The  constant  a  is, 


DILATATION    AND    ITS    PRACTICAL   APPLICATIONS.  315 

of  course,  the  value  of  the  co-efficient  at  40°.     When  more  than  one 
value  is  given  these  refer  to  the  various  principal  dilatations. 

a.  a'. 

Carbon  (retort)       ...        0-00000540  ...  0*0000000144 

Platinum     ......        0*00000905  ...  0*0000000106 

Steel  ......        0-00001095  ...  0-0000000124 

Iron  (compressed)  ...        0-00001188  ...  0'0000000205 

Copper  (native)      ...       0-00001678  ...  0-0000000205 

Silver  ......        0*00001921  ...  0-0000000147 

Lead  ......        0-00002924  ...  0'0000000239 

Ouartz  f  0-00000781  ...  0-0000000205 

10-00001419  ...  0-0000000348 

T    i      -,  (0-00002621  ...  0-0000000160 

(0-00000540  ...  0-0000000087 

(0*00003460  ...  0-0000000337 

Arragonite  ......      j  0*00001719  ...  0-0000000368 

1  0-00001016  ...  0-0000000064 

An  average  value   of  the   co-efficient   of   expansion   of  glass   is 
0-0000085, 

264.  Dilatation  of  Liquids.  —  In  liquids  it  is  merely  the  cubical 
dilatation  with  which  we  have  to  deal.  This  may  readily  be  deter- 
mined if  we  know  the  cubical  dilatation,  K',  of  the  substance  of 
which  the  containing  vessel  is  composed.  Let  K  be  the  unknown 
co-efficient,  and  let  an  amount  of  the  liquid,  of  weight  W,  fill  the 
vessel  (which  must  have  a  narrow  neck)  at  a  temperature,  £,  while 
the  weight  of  the  quantity  which  fills  the  vessel  at  0°  is  W0.  Now 
the  weight  of  the  quantity  which  would  have  filled  it,  had  the  liquid 
been  inexpansible,  is  W0  (1  +  K'£)  ;  but,  since  the  liquid  is  ex- 
pansible, this  amount  is  diminished  in  the  ratio  of  unity  to 
Hence  we  get 


which  determines  K. 

A  very  simple  method,  by  means  of  which  the  co-efficient  of  dilata- 
tion of  a  liquid  may  be  found  without  any  knowledge  of  that  of  the 
substance  of  which  the  containing  vessel  is  composed,  was  devised 
by  Dulong  and  Petit.  In  all  essential  particulars  the  apparatus  con- 
sists of  a  double  U-tube,  abed  (Fig.  154).  The  portion  be  is  occupied 
by  air,  which  separates  the  portions  of  the  liquid  contained  in  ab 
and  cd  while  preserving  continuity  of  pressure  between  them. 
Equal  atmospheric  pressure  acts  at  the  points  a  and  d,  and  the  air 
in  be  is  necessarily  at  uniform  pressure  throughout.  Hence,  when 


316 


A    MANUAL    OF    PHYSICS. 


equilibrium  is  maintained,  the  pressure  per  square  inch  due  to  the 
difference  of  level  ab  must  be  equal  to  the  pressure  per  square  inch 
due  to  the  difference  of  level  dc.  Let  us  suppose  that  the  limb  cd 


d* 


FIG.  154. 


is  raised  to  temperature  t,  while  the  limb  ab  is  kept  at  0°. 
average  expansibility  throughout  the  range  of  temperature  is 

dc-  ab 


The 


If  dv  be  the  increase  of  volume  produced  by  a  small  rise  of  tern 
perature  dt,  while  v,  is  the  total  volume  of  the  liquid  at  0°,  the 


quantity 


dv 

v0dt 


is  usually  taken  as  the  co-efficient  of  dilatation  at  the  temperature  t. 
The  true  co-efficient  at  temperature  t  is  obviously  given  by  the 


ratio 


dv 

vdf 


where  v  is  the  volume  of  the  liquid  at  that  temperature. 

The  values,  given  by  Regnault,  of  the  ordinary  co-efficient  of  dila- 
tation of  mercury  at  temperatures  varying  from  0°  C.  to  850°  C.  are 
well  represented  by  the  formula 

K  -  0-0001791 +0-0000000504*. 

Water  presents  a  marked  peculiarity  as  regards  its  change  of 
volume  with  rise  of  temperature.  Between  its  freezing-point,  0°C., 
and  a  temperature  of  almost  exactly  4°  C.,  its  volume  diminishes  as 
the  temperature  increases.  At  all  higher  temperatures  the  volume 


DILATATION   AND   ITS   PRACTICAL   APPLICATIONS.  317 

increases  as  the  temperature  is  farther  raised.  Thus  water  is  in  a 
condition  of  maximum  density  at  a  temperature  of  about  4°  C. 

The  existence  of  the  temperature  of  maximum  density  is  readily 
shown  by  means  of  Hope's  experiment.  The  necessary  apparatus 
consists  of  a  cylindrical  glass  vessel  the  central  portion  of  which  is 
surrounded  by  a  metal  casing  in  which  a  freezing  mixture  may  be 
placed.  Two  thermometers  are  inserted  horizontally  through  the 
glass  vessel,  one  near  the  top  and  the  other  near  the  bottom,  so  that 
their  bulbs  are  at  the  axis  of  the  cylinder.  The  vessel  being  filled 
with  water,  a  freezing  mixture  is  placed  in  the  casing.  Very  soon 
the  temperature  marked  by  the  lower  thermometer  begins  to 
diminish,  which  shows  that  the  cold  water  is  of  greater  density 
than  the  warm  water  near  the  top  of  the  vessel.  This  goes  on 
until  the  lower  thermometer  registers  a  temperature  of  4°  C.,  at 
which  it  remains.  Soon  afterwards  the  temperature  of  the  water  at 
the  top  of  the  vessel  begins  to  fall,  which  shows  that  the  colder  water 
is  now  ascending  and  must  therefore  be  expanding  ;  and  this  process 
goes  on  until  the  water  at  the  top  freezes  at  0°  C. 

The  maximum-density  point  is  lowered  by  pressure  to  the  extent  of 
about  3°  C.  by  a  pressure  of  one  ton's  weight  per  square  inch. 

Kopp's  determinations  of  the  co-efficient  of  dilatation  of  water  are 
fairly  well  represented  between  0°  C.  and  20°  C.  by  the  formula 


72,000 

The  more  recent  experiments  of  Pierre,  Hagen,  and  Mathiessen 
indicate  that  the  denominator  of  this  fraction  is  about  5£  per  cent, 
teo  large  ;  but  Eossetti's  results  agree  better  with  those  of  Kopp. 

Different  experimenters  have  determined  the  co-efficients  of 
expansion  of  various  liquids  when  under  pressure  sufficient  to  keep 
the  liquid  in  equilibrium  with  its  vapour  at  temperatures  above  the 
ordinary  boiling-point.  Drion  gives  the  subjoined  values  of  the  co- 
efficient of  dilatation  of  sulphurous  acid  : 

Temp.  C.  Co-efficient.  Temp.  C.  Co-efficient. 

0° 0-00173  70° 0-00318 

10° 0-00188  90°     0-00415 

30° 0-00219  110° 0-00592 

50° 0-00259  130° 0-00957 

From  these  results  it  appears  that  the  co-efficient  of  dilatation  of 
this  liquid,  at  about  120°  C.,  is  double  of  that  of  air. 


318  A   MANUAL    OF   PHYSICS. 

Him  gives  the  volume  of  water  at  different  temperatures  as 
follows : 

Temp.'C.  Volume.  Temp.  C.  Volume. 

4°  ...  1-00000                     140°  ...  1-07949 

100°  ...  1-04315                     160°  ...  1-10149 

120°  ...  1-05992                     180°  ...  1-12678 

Consequently,  at  180°  C.,  the  expansibility  of  water  is  nearly  one 
half  of  that  of  air. 

265.  Dilatation  of  Gases. — A  gas  must  be  kept  in  an  enclosed 
space  when  experiments  are  to  be  made  upon  it  with  regard  to  its 
alterations  of,  volume  under  varying  conditions  of  temperature. 
But  we  know  that,  so  long  as  the  temperature  is"  maintained  at  a 
constant  value  the  volume  and  pressure  of  the  gas  are  in  inverse 
proportion  to  each  other.  Hence  we  may  investigate  the  effect  of 
variation  of  temperature  in  two  ways  :  we  may  measure  the  expan- 
sion under  constant  pressure  ;  or,  we  may  measure  the  change  of 
pressure  at  constant  volume. 

By  such  measurements  Charles  (and,  subsequently,  Gay  Lussac) 
was  led  to  the  conclusion  that  the  volume  of  a  given  quantity  of 
any  gas,  under  constant  pressure,  increases  by  a  constant  fraction 
of  its  amount  for  a  given  increment  of  temperature.  This  state- 
ment is  known  as  Charles'  Law,  and  is  represented  symbolically 
in  conjunction  with  Boyle's  Law  by  the  equation 

pv=C  (l  +  oO, 

where  c  and  a  are  constants,  and  the  meaning  of  the  other  quanti- 
ties is  obvious. 

The  pressure  being  kept  constant,  the  volume  increases  by  the 
fraction  a  of  its  amount  at  0°  per  unit  rise  of  temperature ;  a  is 
therefore  the  co-efficient  of  dilatation  under  constant  pressure. 
Again,  if  the  volume  be  kept  constant,  the  pressure  increases  by  the 
fraction  a  of  its  amount  at  0°  per  unit  rise  of  temperature.  Thus 
the  fractional  increase  of  volume 'under  constant  pressure,  and  the 
fractional  increase  of  pressure  at  constant  volume,  have  the  same 
numerical  value  if  the  above  equation  be  rigidly  true. 

The  magnificent  series  of  experiments  carried  out  by  Eegnault 
have  shown  that,  while  the  above  law  is  very  nearly  true  in  the 
cases  of  the  most  permanent  gases,  marked  discrepancies  are  ex- 
hibited when  the  more  readily  liquefiable  gases  are  employed.  His 
experiments  on  the  undernoted  gases  gave  as  the  value  of  the  dilata- 
tion between  0°  C.  and  100°  C. : 


DILATATION   AND   ITS   PRACTICAL   APPLICATIONS.  319 

Hydrogen         0-3667  0-3661 

Air         0-3665  ...         ...  0'3670 

Nitrogen  0'3668  0'3670 

Carbonic  oxide  ...  0'3667  '  0*3669 

Carbonic  acid 0'3688  0'3710 

Cyanogen         0-3829  0-3877 

Sulphurous  acid          ...  0'3845  0-3903 

The  figures  in  the  second  column  represent  the  value  of  the  co- 
efficient as  determined  at  constant  volume;  those  in  the  third 
column  are  the  values  determined  under  constant  (atmospheric) 
pressure.  With  a  single  exception  (in  the  case  of  hydrogen)  the 
latter  number  exceeds  the  former. 

When  the  initial  pressure  of  gases  at  0°  is  increased,  the  dilata- 
tion between  0°  and  100°  increases.  Some  of  Regnault's  results  for 
air  are : 

109-72         ...         149-31         ...         0-3648 

374-67         ...         510-35         ...         0-3659 

760-00         ...       1038-54         ...         0'3665 

1692-53         ...       2306-23         ...         0-3680 

3655-66         ...       4992-09         ...         0'3709 

The  numbers  in  the  first  two  columns  represent  respectively  the 
pressure  per  unit  area  at  0°  and  at  100°  expressed  in  terms  of  the 
weight  at  0°  of  a  column  of  mercury  of  unit  section  and  one  milli- 
metre in  height. 

For  carbonic  acid  Begnault  gives  the  similar  results  : 

758-5  ...  1034-5  ...  0-36856 

901-1  ...  1230-4  ...  0-36943 

1742-9  ...  2387-7  ...  0-37523 

3589-1  ...  4759-0  ...  0-38598 

The  variation  in  the  case  of  this  gas  is  therefore  greater  than  in 
the  case  of  air. 

The  results,  for  the  same  two  gases,  under  constant  pressure, 
are  : 

Air 760  ...       0-36706 

2525  ...       0-36944 

Carbonic  acid       ...         760  ...       0-37099 

2520  ...       0-38455 

Regnault's  apparatus  consisted  essentially  of  a  glass  bulb  D 
(Fig.  155),  which  contained  the  gas  and  communicated  through 
the  tube  E  with  a  reservoir  AB  which  was  filled  with  mercury. 


320 


A    MANUAL    OF    PHYSICS. 


A  tube  F,  open  to  ths  atmosphere,  also  communicated  with  the 
reservoir.  When  the  co-efficient  was  to  be  determined  at  constant 
volume,  a  plug  C  was  screwed  in  until  the  mercury  stood  at  the 
points  E  and  F  in  the  tubes — E  being  a  fixed  point.  D  was,  under 
these  conditions,  surrounded  successively  by  melting  ice,  and  by  the 
steam  from  boiling  water,  and  the  pressure  in  each  case  was  found 


A  B 

FIG.  155. 

from  the  difference  of  level  of  the  mercury  in  the  two  tubes  and 
the  known  barometric  pressure.  Suitable  corrections  had,  of  course, 
to  be  applied  for  the  expansion  of  the  bulb  by  heating  or  by  pressure. 

When  the  dilatation  was  observed  under  constant  pressure,  the 
plug  C  was,  at  each  temperature,  screwed  out  until  the  mercury 
stood  at  the  same  level  in  the  two  tubes.  The  pressure  was  then 
equal  to  that  of  the  atmosphere. 

Other  volumetric,  or  gravimetric,  methods  have  been  employed 
by  Regnault  and  various  experimenters. 

266.  Absolute  Zero  of  Temperature. — We  may  write  the  equa- 
tion which  expresses  Boyle's  and  Charles'  Laws  in  the  form 


pv- 


Now  as  the  second  term  within  the  brackets  represents  temperature, 
the  first  term  must  also  represent  a  temperature,  for  the  dimen- 
sions (§  27)  of  all  the  terms  of  a  physical  equation  must  be  iden- 
tical. And  the  numerical  value  of  that  temperature  is  about  273, 
since  a  =  0*003665.  Hence  we  must  suppose  that  the  zero  of  the 
Centigrade  scale  corresponds  to  a  temperature  of  273  degrees  on  this 


DILATATION   AND    ITS    PRACTICAL   APPLICATIONS.  321 

new  scale,  which  we  may  call  a  scale  of  absolute  temperature,  since 
its  magnitude  is  independent  of  the  particular  gas  employed. 

To  look  at  the  matter  from  another  point  of  view,  we  observe  that, 
as  t  diminishes  to  zero  and  then  becomes  an  increasing  negative 
quantity,  the  product  pv  constantly  diminishes,  and  finally  becomes 
zero  when  t  =  -  273°.  If  the  volume  is  constant,  this  means  that 
the  pressure  vanishes  when  t  has  this  value.  But  the  pressure  of  a 
gas  is  due  to  the  motion  of  its  particles,  and  hence,  when  t=  -  273°, 
the  particles  cease  to  move,  and  the  gas  is  therefore  totally  deprived 
of  heat. 

The  above  equation  may  therefore  be  written  in  the  form 

pv  =  IU, 
where  R  =  Ca  and  t  represents  absolute  temperature. 

We  shall  see  later  (§  303)  that  this  estimate  of  the  position  of  the 
absolute  zero  on  the  Centigrade  scale  is  confirmed  by  thermo- 
dynamical  considerations. 

267.  Measurement  of  Temperature. — The  most  usual  method 
of  measuring  temperature  is  by  means  of  the  expansion  of  a  liquid 
or  a  gas.  Mercury  is  generally  used  in  the  former  case,  air  in  the 
latter. 

A  glass  tube  of  narrow,  and  as  nearly  as  possible  uniform,  bore  is 
first  chosen.  If  the  bore  be  not  quite  uniform,  its  variations  are 
determined  by  the  process  of  calibration.  This  consists  in  running 
a  small  quantity  of  mercury  along  the  tube  from  one  e*nd  to  the 
other,  and  measuring  its  length  at  the  various  parts.  The  quotient 
of  the  weight  of  the  mercury  by  its  specific  gravity  and  th«  length 
of  the  column  at  any  part  gives  the  mean  section  of  that  portion  of 
the  tube.  Next,  a  bulb  is  blown,  on  one  end  of  the  tube,  of  a  size 
which  is  determined  by  the  bore  of  the  stem,  the  expansibility  of  the 
liquid  to  be  used,  and  the  required  length  of  a  scale  division. 
The  bulb  is  now  slightly  heated  to  expel  some  air,  and  the  instru- 
ment is  then  inverted  in  a  vessel  of  the  liquid  with  which  it  is  to  be 
filled.  As  the  bulb  cools,  some  of  the  liquid  enters  it.  This  liquid 
is  then  boiled  in  the  bulb,  and  its  vapour  expels  the  remaining  air. 
A  repetition  of  the  process  of  inversion  of  the  bulb  and  stem  in  the 
liquid  will  result  in  both  being  entirely  filled.  So  much  of  the 
liquid  is  then  run  out  that  the  remainder  scarcely  fills  the  stem 
when  it  is  boiled  once  more.  The  vapour  drives  out  the  air  which 
entered,  and  the  tube  is  then  hermetically  sealed. 

Two  fixed  points  must  now  be  determined  on  the  stem.     As  we 
shall  see  afterwards  (§§  273,  275),  this  may  be  done  by  means  of 
melting  ice,  and  of  the  steam  coming  from  boiling  water  in  a  nearly"  • 
closed  vessel. 

21 


322  A   MANUAL    OF    PHYSICS, 

But,  before  these  points  are  determined,  a  considerable  time 
should  be  allowed  to  elapse,  for  the  bulb  will  not  shrink  quickly  to 
its  final  volume.  The  process  of  shrinkage  usually  goes  on  for 
years,  though,  by  careful  annealing,  the  effect  may  be  considerably 
lessened. 

To  determine  the  lower  fixed  point  of  the  scale,  the  bulb  and  part 
of  the  stem  are  surrounded  by  melting  ice.  The  final  position  of 
the  extremity  of  the  liquid  column  is  marked  on  the  stem. 

The  upper  fixed  point  is  determined  by  surrounding  the  bulb,  and 
as  much  of  the  stem  as  possible,  by  the  steam  which  is  escaping 
from  water  boiling  under  a  pressure  of  one  atmosphere.  The  final 
position  of  the  liquid  is  then  marked  on  the  stem. 

The  distance  between  the  two  marked  points  is  then  divided  into 
a  number  of  equal  parts. 

On  the  Centigrade  seals,  the  lower  fixed  point  is  marked  0°,  and 
the  higher  is  marked  100° ;  on  the  Fahrenheit  scale  the  lower  is 
marked  32°,  and  the  higher  212°.  Thus,  on  the  Centigrade  scale, 
there  are'100  divisions  between  the  boiling-point  and  the  freezing- 
point  of  water — hence  its  name ;  on  the  Fahrenheit  scale  there  are 
180  divisions  between  these  points.  The  Fahrenheit  zero  was 
determined  by  a  freezing-mixture  of  snow  and  salt,  which  gave  the 
lowest  temperature  known  at  the  time  when  Fahrenheit  first  con- 
structed his  thermometers.  The  relation  between  the  two  scales  is 
obviously  given  by  the  equation 

F-32^   C 
180   "100* 

where  F  and  C  respectively  represent  the  Fahrenheit  and  Centigrade 
scale  readings. 

The  same  interval  on  Reaumur's  scale  is  divided  into  80  equal 
parts,  the  zero  being  the  same  as  that  of  the  Centigrade  scale. 

As  already  remarked,  the  liquid  generally  used  is  mercury.  For 
the  measurement  of  temperatures  below  the  freezing  -  point  of 
mercury,  alcohol  is  employed. 

The  air  thermometer  is  of  great  use  in  the  determination  of 
temperatures  above  those  at  which  mercury  can  be  employed  ;  and 
its  readings  agree  pretty  closely  with  those  of  the  true  absolute 
scale  of  temperature  as  determined  by  thermodynamical  considera- 
tions. The  following  numbers,  taken  from  -Regnault's  results,  show 
the  difference  between  the  Centigrade  scale  of  the  air  thermometer 
and  that  of  the  mercury  thermometer : 

Air  ...         ...  0°  20°       406        60°       80°        100°  200°       300° 

Mercury      ...  0°  19o;98  396<67  59°-62  79°'7S  100°  202°-78  308°'34. 


DILATATION    AND    ITS    PRACTICAL    APPLICATIONS.  323 

Self-registering  thermometers  are  frequently  employed  for  the 
purpose  of  indicating  the  maximum,  or  the  minimum,  temperature 
attained  between  two  given  periods.  In  the  usual  form  of  the 
maximum  thermometer,  the  expanding  column  of  mercury  pushes 
an  iron  index  along  the  tube.  This  index  is  left  behind  when  the 
column  contracts,  for  mercury  does  not  wet  iron.  In  the  usual 
minimum  thermometer,  alcohol  is  used  along  with  a  glass  index. 
The  liquid,  when  expanding,  flows  past  the  index  ;  when  it  contracts, 
it  pulls  the  index  with  it,  for  its  surface  tends  to  take  the  smallest 
possible  area  (§  120),  and  this  is  attained  when  it  occupies  the  space 
between  the  index  and  the  walls  of  the  tube. 

Continuously  -  registering  thermometers  are  also  used.  The 
principle  of  one  of  the  best  forms  of  these  instruments  is  identical 
with  that  of  the  Bourdon  gauge.  Let  abed  (Fig.  156)  be  a  longitudinal 
section  of  a  hollow  metal  receiver  in  its  unstrained  state,  and  let  us 
suppose  that  the  receiver  is  filled  with  a  liquid.  As  the  temperature 


FIG.  156. 

rises,  the  pressure  increases  ;  and,  since  cd  is  greater  than  ab,  the 
sum  of  the  moments  of  the  forces  tending  to  straighten  the  receiver 
exceeds  the  sum  of  those  which  act  contrariwise.  A  system  of 
levers  attached  to  such  a  receiver  (fixed  at  one  end)  traces  a  con- 
tinuous record  on  a  properly-graduated  paper  placed  on  a  drum 
which  revolves  slowly  at  a  uniform  rate. 

Pyrometers  are  used  for  the  rough  determination  of  very  high 
temperatures.  In  Daniell's  pyrometer,  a  bar  of  platinum  is  slipped 
into  a  hole  bored  in  a  rod  of  graphite.  A  plug  of  graphite,  or 
baked  clay,  rests  on  the  top  of  the  bar  and  fits  tightly  into  the  hole, 
or  is  otherwise  kept  tightly  in  position.  When  the  platinum  (which 
rests  on  the  bottom  of  the  hole)  expands,  the  plug  is  pushed  out, 
and  remains  in  its  position  of  maximum  displacement  when  the 
temperature  falls.  From  the  increase  of  length  of  the  platinum 
thus  registered,  we  can  calculate  the  temperature  of  any  furnace 
in  which  it  may  be  placed,  on  the  assumption  that  the  law  of 
dilatation,  determined  throughout  moderate  ranges  of  temperature, 
holds  up  to  the  high  temperatures  of  the  furnace. 

Other  methods  are  also  employed  for  the  determination  of  high 
temperatures.  See  §§  269,  343. 

21—2 


CHAPTER  XXIII. 

EFFECTS  OF  THE  ABSORPTION  OF  HEAT  :  CHANGE  OF  TEMPERATURE 
AND  CHANGE  OF  STATE. 

268.  Unit  of  Heat.  Specific  Heat.  Thermal  Capacity. — One  of 
the  most  marked  effects  of  absorption  of  heat  is  a  rise  of  the  tem- 
perature of  the  heated  body.  In  some  cases,  no  change  of  temperature 
takes  place,  and  the  effect  which  appears  instead  is  a  change  of  the 
physical  state  of  the  substance.  But,  before  discussing  these  effects, 
we  must  consider  the  methods  of  measuring  the  amount  of  heat 
required  to  produce  a  given  change ;  and  this,  in  turn,  necessitates 
the  adoption  of  a  definite  unit  of  heat. 

We  may  conveniently  adopt  as  our  unit  the  quantity  of  heat 
which  is  required  to  raise  the  temperature  of  one  pound  of  ice- 
cold  water  to  1°  C. 

A  given  quantity  of  heat  may  therefore  be  measured  by  the 
number  of  pounds  of  ice-cold  water  which  it  can  raise  in  tempera- 
ture to  1°  C.  We  might  measure  it  also  by  the  number  of  pounds  of 
ice  at  0°  C.  which  it  is  just  able  to  melt ;  for,  as  we  shall  see  shortly, 
the  number  of  units  of  heat  which  are  required  to  just  melt  one 
pound  of  ice  at  0°  C.  is  quite  definite  and  measurable.  More 
generally,  we  might  adopt  as  our  unit  the  amount  of  heat  necessary 
to  produce  any  definite  physical  change,  and  then  we  might  deter- 
mine what  fraction  of  the  unknown  quantity  of  heat  could  produce 
this  change.  The  number  of  units  in  the  unknown  quantity  would 
be  the  reciprocal  of  this  fraction. 

We  define  the  specific  heat  ;of  a  substance,  under  given  condi- 
tions, as  the  quantity  of  heat  which  is  required  to  raise  the  tem- 
perature of  one  pound  of  the  substance  by  1°  C.  From  this 
definition,  and  from  our  previous  definition  of  the  unit  of  heat, 
it  follows  that  we  must  consider  the  specific  heat  of  one  pound  of 
water 'at  0°  C.  to  be  unity. 

More  strictly,  we  should  define  it  as  the  rate  at  which  the  quantity 


CHANGE    OF    TEMPERATURE    AND    CHANGE    OF   STATE.  325 

of  heat  supplied  to  the  substance,  per  pound  of  its  mass,  varies  with 
the  temperature.  But,  in  all  actual  cases,  there  is  no  practical 
difference  between  the  two  definitions. 

The  mean,  or  average,  specific  heat  of  a  substance,  throughout  a 
given  range  of  temperature,  is  obtained  by  dividing  the  amount 
of  heat  which  is  required  to  raise  one  pound  of  the  substance 
through  the  given  range  by  the  difference  between  the  two  extreme 
temperatures. 

The  Thermal  Capacity  of  a  substance  is  the  quantity  of  heat 
which  is  required  to  raise  the  temperature  of  unit  volume  of  the 
substance  by  one  degree.  It  is  therefore  equal  to  the  product  of 
the  specific  heat  and  the  density  of  the  substance. 

269.  Specific  Heat  of  Solids  and  Liquids. — Various  methods 
are  used  for  the  determination  of  specific  heat. 

In  one  method,  use  is  made  of  the  fact  that  the  rate  of  emission 
of  heat  from  a  body  at  a  given  temperature  depends  only  upon  the 
nature  of  its  surface.  Hence,  if  we  fill  a  thin  metal  globe  successively 
with  two  different  liquids,  and  observe  the  rate  of  cooling  of  each 
liquid  at  the  same  temperature,  we  can  compare  the  specific  heats  of 
the  two  liquids.  For,  if  m,  s,  r,  and  m',  s',  r',  represent  respectively 
the  mass,  the  specific,  and  the  rate  of  cooling  of  the  liquids,  we  have 

msr=mrs'rf. 
If  one  of  the  liquids  be  water,  so  that  s  =  l,  we  get 


In  actual  experiment,  the  liquids  would  be  raised  to  a  common 
high  temperature  and  readings  of  their  temperatures  would  be 
taken  at  equal  intervals  of  time  as  they  cooled.  If  a  curve  were 
then  drawn  the  ordinates  of  which  represented  temperature,  while 
the  abscissas  represented  time,  the  rate  of  cooling  would  be  given 
by  means  of  the  tangent  to  the  curve.  Thus  (Fig.  157),  to  find 
the  rate  of  cooling  at  a  temperature  9,  draw  the  tangent  ab  to 
the  curve  at  the  point  P,  corresponding  to  9,  and  let  it  intersect  the 
time-axis  in  the  point  a  and  the  temperature-axis  at  the  point  b ; 
the  rate  of  cooling  is  ob/oa. 

Another  method  of  determining  specific  heat  is  known  as  the 
Method  of  Mixture.  Let  m  pounds  of  one  substance  at  tempera- 
ture t  be  mixed  with  m'  pounds  of  another  at  temperature  t',  and 
let  the  specific  heats  of  the  two  substances  be  respectively  s  and  sr. 
If  the  temperature  of  the  mixture  be  9,  the  heat  lost  by  the  hotter 
body  (say  that  of  mass  m')  is  m's'(t'  -  9).  Similarly,  the  heat  gained 


326 


A   MANUAL    OF    PHYSICS. 


by  the  colder  substance  is  ms(9  -  t).  A]so,  if  ^  be  the  mass  of  the 
vessel  which  contains  the  substances  (and  we  may  regard  the 
stirring-rod  used  to  mix  the  substances  as  forming  part  of  it)  while 
<r  represents  its  specific  heat,  the  heat  given  to  the  vessel  is  ^(O  -  t). 
Here  we  assume  that  the  colder  substance  was  originally  contained 
in  the  vessel,  the  hotter  substance  being  introduced  from  without. 


FIG.  157. 

Instead  of  //o-  we  may  write  w — a  unit  multiplier  being  under- 
stood. The  meaning  of  this  is  that  w=^o  is  the  number  of  pounds 
of  water  (specific  heat  =  unity)  which  require  the  same  supply  of 
heat  to  produce  the  rise  of  temperature  (9  -  t)  as  the  vessel  required. 
The  quantity  \ia  is  therefore  called  the  water-equivalent  of  the  vessel. 

We  mav  therefore  write 


provided  that  the  cold  substance  is  water.  If  the  value  of  w  be  known, 
this  equation  enables  us  to  find  the  value  of  s'.  If  w  be  not  known, 
a  second  experiment  in  which  the  mass,  m,  of  water  is  varied,  will 
lead  to  another  similar  equation  by  means  of  which  w  may  be 
eliminated  or  determined.  Of  course,  the  value  of  w  may  be  found 
by  one  experiment  alone,  in  which  two  quantities  of  water,  at 
different  temperatures,  are  mixed. 

In  an  accurate  experiment  of  this  kind,  precautions  are  taken 
that  there  shall  be  as  little  loss  of  heat  by  radiation  as  possible. 
This  may  be  effected  by  making  the  vessel  which  contains  the 
water  (or  other  liquid)  of  a  substance  which  is  a  bad  radiator  of 
heat ;  and,  in  addition,  this  vessel  is  placed  inside  a  second  similar 
vessel,  contact  between  the  two  being  prevented  by  means  of  bodies 
which  do  not  readily  conduct  heat.  If  necessary,  a  third  vessel 
may  be  used  ;  and  then  a  correction  may  be  made  for  the  slight 
amount  of  heat  which  is  still  lost  by  radiation. 


CHANGE    OF    TEMPERATURE    AND    OHANGE    OF    STATE.  327 

A  third  method  of  determining  specific  heat  is  by  the  Fusion  of 
Ice.  As  we  shall  see  subsequently  (§  274),  a  definite  amount  of 
heat  is  required  to  just  melt  one  pound  of  ice.  Let  H  be  this 
quantity.  If  M  pounds  of  a  substance  melt  m  pounds  of  ice  in  the 
process  of  cooling  from  T°  C.  to  0°  C.,  the  average  specific  heat,  S, 
throughout  that  range  of  temperature  is  given  by  the  equation 

wH  =  MST. 

Bunsen  and  others  have  used  forms  of  apparatus  in  which  the 
quantity  m  is  found  by  means  of  the  decrease  of  volume  of  a  mixture 
of  ice  and  water  when  the  heat  which  is  given  out  by  the  cooling 
body  melts  some  of  the  ice. 

In  general  the  specific  heat  of  a  substance  increases  with  rise  of 
temperature.  But  the  specific  heat  of  platinum  varies  very  slightly 
with  temperature,  so  that  the  range  of  temperature  through  which 
a  mass  of  that  metal  cools  is  closely  proportional  to  the  quantity  of 
heat  which  it  emits.  This  fact  is  utilised  in  the  measurement  of 
high  temperatures. 

Regnault  found  that  the  quantity  of  heat  required  to  raise  the 
temperature  of  a  pound  of  water  from  0°  C.  to  t°  C.  is  represented  by 
the  equation 

H  =  t  +  0-00002**  +  0-0000003*8, 

and  that  therefore  the  true  specific  heat,  at  any  temperature,  is 
given  by 

^r  =  l  +  0-00004*  +  0-0000009*2. 

His  experiments  were  carried  out  at  various  temperatures  between 
0°  C.  and  230°  C. 

The  specific  heat  of  ice  is  almost  exactly  equal  to  0'5,  and 
Regnault  found  that  it  is  diminished  by  decrease  of  temperature. 

This  table  gives  the  specific  heat  of  various  elementary  substances 
at  ordinary  temperatures  : 

Solid.  Liquid. 

Water  0-500  (at  0°  C.)  ...1-000 

Glass  0-180 

Iron  0-114 

Copper  0-096 

Zinc  0-094 

Silver  0'057 

Tin  0-056 0'064 

Mercury 0-031     0'033 

Lead  0-031  ..   0'040 


328  A    MANUAL    OF    PHYSICS. 

It  is  worthy  of  notice  that  the  specific  heat  of  water  is  consider- 
ably in  excess  of  that  of  any  of  the  other  substances ;  and  that,  in 
general,  the  specific  heat  of  any  substance  in  the  liquid  state  exceeds 
that  of  the  same  substance  in  the  solid  state. 

270.  Law  of  Dulong  and  Petit. — Dulong  and  Petit  found  that 
the  product  of  the  specific  heat  of  any  elementary  solid  into  its 
atomic  weight  is  practically  constant.  An  alternative  statement  is 
that  the  water -equivalent  of  an  atom  of  each  elementary  solid  is 
practically  constant.  The  numbers  in  the  first  column  below  repre- 
sent specific  heat ;  those  in  the  second  column  represent  atomic 
weight ;  and  those  in  the  third  give  the  product  of  these  two  quantities. 

Iron     0-114  54-5  6-2 

Copper            ...  0-096  63'5  6-1 

Zinc    ...         ...  0-094  ...         ...  64'5  6-1 

Silver 0'057  108-0  6-2 

Tin      0-056  118-0  6-6 

Mercury  (liquid)  0'033  202-0  6-6 

Lead 0'031  207*0  6-4 

Similar  results  have  been  established  for  various  series  of 
chemical  compounds.  The  value  of  the  constant  varies  from  one 
such  series  to  another. 

271.  Specific  Heat  of  Gases  and  Vapours. — The  specific  heat  of  a 
gas,  at  any  given  temperature,  may  be  measured  in  two  different 
ways.  We  may  keep  the  pressure  constant,  or  we  may  keep  the 
volume  constant.  The  numerical  values  of  the  specific  heat,  as 
obtained  by  the  two  methods,  are  different ;  and  so  we  speak  of  the 
Specific  Heat  at  Constant  Pressure  and  the  Specific  Heat  at  Con- 
stant Volume. 

The  experimental  difficulties  which  are  encountered  in  the  deter- 
mination of  the  specific  heat  at  constant  volume  are  almost  insur- 
mountable ;  but,  in  the  case  of  approximately  perfect  gases  [which 
closely  obey  the  law  pv  =  ~Rt  (§  266)] ,  the  principles  of  thermodyna- 
mics show  (§  301)  that  the  difference  of  the  two  specific  heats  is 
equal  to  the  quantity  E.  Hence  it  is  only  necessary  to  measure  the 
specific  heat  of  such  substances  at  constant  pressure.  The  method 
which  is  adopted  consists  in  passing  a  slow  stream  of  the  gas,  under 
constant  pressure,  through  two  spiral  tubes,  in  the  first  of  which  its 
temperature  is  raised  to  a  known  amount,  while  in  the  second  it  is 
lowered  to  a  known  amount.  The  amount  of  heat  which  is  given 
out  in  the  process  of  cooling  is  measured  by  the  rise  of  temperature 
of  a  known  mass  of  water,  and  the  mass  of  the  gas  which  gives  out 
the  heat  is  determined  from  a  measurement  of  its  volume. 


CHANGE    OF    TEMPERATURE    AND    CHANGE    OF    STATE.  329 

Delaroche  and  Berard,  who  first  used  the  above  method,  were  led 
to  believe  that  the  specific  heat  of  a  gas  varies  with  its  pressure. 
Regnault,  who  worked  with  improved  apparatus,  found  that  it  is 
independent  of  the  pressure — not  merely  in  the  case  of  a  gas  such  as 
air  which  sensibly  obeys  Boyle's  law,  but  also  in  the  cases  of  car- 
bonic acid  and  hydrogen.  He  found  also  that  while  the  specific 
heat  of  carbonic  acid  increases  markedly  as  the  temperature  rises, 
the  specific  heat  of  air  is  independent  of  the  temperature.  It  is, 
therefore,  by  Boyle's  and  Charles'  laws,  independent  of  the  volume. 
And  we  may  in  all  probability  conclude  that  the  specific  heats  of  all 
ga*ses  which  closely  obey  Boyle's  law  are  absolutely  constant.  It 
follows  that  the  thermal  capacity  of  such  a  gas  is  proportional  to  its 
density.  The  results  in  the  table  are  due  to  Eegnault. 

Specific  Heat  of  Simple  Gases. 

Hydrogen       3'4090        Oxygen     0'2175 

Nitrogen         0'2438        Chlorine 0-1210 

Air       0-2374        Bromine 0'0555 

Specific  Heat  of  Compound  Gases. 

Ammonia       0-5084        Carbonic  acid       ...     0-2169 

Carbonic  oxide          ...     0-2450       Hydrochloric  acid       0-1852 
Sulphuretted  hydrogen    0-2432        Sulphurous  acid  ...     0-1544 

The  ratio  of  the  two  specific  heats  of  a  gas  may  be  found  from  the 
speed  of  sound  in  that  gas  (see  §  158).  In  air  and  some  other  gases 
the  specific  heat  at  constant  pressure  is  almost  exactly  1*4  times 
greater  than  that  at  constant  volume.  Jamin  and  Richard  have 
obtained,  by  a  direct  experimental  method  (in  which  the  tempera- 
ture of  the  gas  is  raised  by  means  of  a  known  amount  of  heat 
developed  by. the  passage  of  an  electric  current  through  a  metallic 
wire),  results  which  agree  well  with  those  obtained  by  the  acoustic 
method. 

The  specific  heat  of  water-vapour,  under  constant  pressure,  is 
about  0-48. 

The  specific  heat  of  ,some  saturated  vapours — for  example,  those 
of  water  and  carbon  bisulphide — is  negative.  Such  vapours,  no  liquid 
being  present,  become  superheated  under  increase  of  pressure  unless 
heat  be  withdrawn  from  them  :  and,  under  decrease  of  pressure,  they 
will  condense  unless  heat  be  supplied  to  them.  Their  specific  heat 
diminishes  in  numerical  magnitude  as  the  temperature  is  raised. 
On  the  other  hand,  the  specific  heat  of  the  saturated  vapour  of  ether 
is  positive,  and  increases  with  increase  of  temperature.  In  all  cases 


330  A    MANUAL    OF    PHYSICS. 

the  actual  increase  of  specific  heat  is  positive.     In  benzine  this 
increase  results  in  a  change  of  sign  of  the  specific  heat. 

272.  Change  of  Molecular  State.  Latent  Heat. — We  have 
already  remarked  that,  in  some  cases,  the  application  of  heat  to  a 
body  does  not  produce  a  rise  of  temperature,  and  that  a  change  of 
molecular  condition  appears  instead. 

Thus  the  application  of  heat  to  ice  below  0°  C.,  raises  its  tempera- 
ture and  causes  it  to  expand.  When  the  ice  reaches  the  temperature 
of  0°  C.,  melting  takes  place  continuously  as  more  and  more  heat  is 
applied.  When  the  melting  is  complete  the  temperature  again  rises 
until,  under  ordinary  atmospheric  conditions,  the  water  boils  at 
100°  C.  When  alFlhe  liquid  has  boiled  away  the  temperature  again 
rises  until  the  water  begins  to  break  up  into  its  constituent 
elements. 

If,  at  any  stage  of  the  above  process,  the  application  of  heat  were 
stopped,  and  heat  were  withdrawn  instead,  the  various  changes 
would  be  gone  through  in  precisely  the  reverse  order. 

The  change  from  the  solid  state  to  the  liquid  state  is  termed  the 
process  of  melting  or  of  fusion.  The  reverse  process  is  called  solidifi- 
cation or  r  eg  elation. 

The  change  from  the  liquid  condition  to  the  state  of  vapour  is 
known  as  vaporisation,  and  the  direct  change  from  the  solid  state 
to  the  state  of  vapour  is  called  sublimation. 

In  no  case  probably  do  these  changes  take  place  suddenly. 
Evaporation  may  go  on  at  all  temperatures;  and  many  solids 
gradually  soften  before  they  melt.  It  is  most  probable  that  such 
softening  occurs  even  in  the  case  of  ice  and  similar  bodies  which 
appear  to  melt  suddenly. 

The  heat  which  is  applied  in  order  to  produce  fusion  or  vaporisa- 
tion, without  change  of  temperature,  was  called  Latent  Heat  because 
it  does  not  give  rise  to  effects  which  can  be  measured  by  any  ordinary 
thermometric  apparatus. 

273.  Fusion  and  Solidification. — The  laws  which  regulate  the 
process  of  fusion  and  which  have  already  been  alluded  to  in  the  case 
of  water,  may  be  enunciated  as  follows  : 

1.  So  long  as  the  pressure  is  maintained  constant  there  is  a 
definite  melting-point  for  every  solid ; 

2.  If  the  solid  and  the  liquid  be  well  mixed,  and  heat  be  applied 
slowly,  the  temperature  of  the  mixture  remains  at  the  melting- 
point  until  the  whole  of  the  solid  has  melted. 

In  the  statement  of  the  first  law  the  condition  of  constant 
pressure  is  imposed.  Only  in  the  case  of  a  substance,  the  volumes 
of  equal  masses  of  which,  in  the  solid  and  liquid  conditions 


CHANGE    OF    TEMPERATURE    AND    CHANGE    OF    STATE.  831 

respectively,  were  equal,  however  the  pressure  might  vary,  would 
the  restriction  be  unnecessary. 

If  a  given  liquid,  such  as  water,  expands  in  the  act  of  solidifica- 
tion, the  application  of  pressure  will  tend  to  prevent  the  solidification, 
because  it  tends  to  prevent  expansion.  Consequently,  more  heat 
must  be  withdrawn  from  the  liquid  in  order  that  the  change  of  state 
may  be  brought  about.  But  this  implies  that  the  temperature  is 
lowered. 

Similarly,  the  melting-point  (or  rather,  from  our  present  point  of 
view,  the  solidifying-point)  of  a  substance,  such  as  paraffin,  which 
contracts  in  the  act  of  solidification,  is  raised  by  the  application  of 
pressure.  For  the  application  of  pressure  makes  the  change  occur 
more  readily  ;  and.  consequently,  less  heat  has  to  be  abstracted  in 
order  that  the  action  may  proceed.  In  other  words,  the  tempera- 
ture at  which  the  change  occurs  is  raised. 

The  theoretical  investigation  of  the  problem  will  be  given  later 
(§  300). 

Professor  James  Thomson  predicted  from  theory  that  the  melting- 
point  of  ice  would  be  lowered  by  pressure  to  the  extent  of  00<0075  C. 
per  atmosphere  of  pressure.  This  prediction  was  fully  verified  by 
Sir  W.  Thomson.  The  second  column  in  the  table  below  gives  the 
melting-points  of  paraffin,  in  Centigrade  degrees,  which  correspond  to 
the  pressures,  in  atmospheres,  which  are  given  in  the  first  column. 
These  results  were  obtained  by  Bunsen.  The  second  and  third 
pairs  of  columns  give  similar  results,  obtained  by  Hopkins,  for 
stearine  and  for  sulphur  respectively. 

1     ...     46°'3     ...         1     ...     72°-5     ...         1     ...     107° 

85     ...     48°-9     ...     519     ...     73°'6     ...     519     ...     135° 

100     ...     49°-9     ...     792     ...     79°'2     ...     792     ...     140° 

The  motion  of  glaciers  is  due,  in  large  part  at  least,  to  the  fact 
that  the  melting-point  of  ice  is  lowered  by  pressure.  When  the 
pressure  arising  from  the  weight  of  the  superincumbent  strata  of  ice 
increases  to  sc  sufficient  extent  at  any  point  in  the  bed  of  the  glacier, 
liquefaction  takes  place,  and  the  water  flows  round  the  obstacle  to 
the  presence  of  which  the  increase  of  pressure  was  due.  But,  the 
pressure  being  relieved  at  the  given  point  (and,  therefore,  handed 
on  to  another  part  of  the  mass)  because  of  the  contraction  which 
takes  place  in  melting,  the  water  again  becomes  solid :  and  so  the 
glacier,  by  a  continuous  process  of  melting  and  re-solidification, 
gradually  moves  down  the  valley  which  it  occupies. 

For  the  same  reason,  snow  which  is  not  too  cold  may  be  readily 
kneaded  into  a  compact  mass  of  ice  :  and  a  wire,  which  is  loaded  at 


Od'^  A   MANUAL    OF    PHYSICS. 

its  two  extremities,  and  is  hung  over  a  bar  of  ice,  will  gradually  cut 
its  way  through  the  bar  without  actually  dividing  it  into  two  parts ; 
for,  though  the  ice  below  the  wire  is  melted  by  the  pressure,  the 
water  which  is  produced  flows  round  the  wire  and  solidifies  above 
it.  The  path  of  the  wire  through  the  clear  ice  can  be  readily  traced 
by  means  of  the  air-bubbles  which  the  ice  contains. 

Sir  W.  Thomson  has  found  that  the  earth  as  a  whole  is  more 
rigid  than  an  equal  globe  of  glass.  This  could  be  explained  if  the 
melting-point  of  the  average  materials  of  the  earth  is  raised  by 
pressure.  It  is  well  known  that  this  is  so  in  the  case  of  ordinary 
lavas. 

Under  special  circumstances,  the  laws  of  fusion,  as  enunciated 
above,  may  be  violated. 

Thus  Fahrenheit  showed  that  water  which  completely  fills  a 
closed  glass  vessel  may  be  cooled  below  0°  C.  before  it  freezes.  And 
Gay-Lussac  showed  that  the  temperature  may  be  lowered  to  -  12°  C. 
in  an  open  glass  vessel  if  the  surface  of  the  water  be  protected  from 
the  air  by  a  layer  of  oil.  The  same  phenomenon  appears  in  the 
cases  of  other  liquids,  such  as  melted  tin,  phosphorus,  and  sulphur. 
In  all  such  cases  any  vibration  of  the  liquid  must  be  avoided,  or 
solidification  will  take  place  suddenly. 

The  melting-points  of  different  substances  vary  greatly.  On  the 
one  hand,  hydrogen  can  only  be  solidified  by  the  aid  of  powerful 
freezing  mixtures;  and,  on  the  other  hand,  gas-coke  can  only  be 
softened  at  the  temperature  of  the  electric  arc. 

Table  of  Melting  points. 

Mercury -40°         Sulphur         115° 

Ice          0°         Zinc ...       415° 

Phosphorus       ...         44°        Wrought  iron          ...  1500°  (?) 

The  melting-point  depends  upon  the  purity  of  the  substance. 
Thus  the  melting-points  of  different  alloys  of  the  same  two  sub- 
stances vary  greatly. 

274.  Latent  Heat  of  Fusion. — The  latent  heat  of  fusion  of  any 
substance  may  be  defined  as  the  quantity  of  heat  which  is  required 
to  just  melt  one  pound  of  that  substance  at  its  ordinary  tempera- 
ture of  fusion. 

This  latent  heat  is  given  out  again  on  re-solidification.  Its 
amount,  for  each  definite  substance,  under  given  conditions  of 
pressure,  is  invariable. 

The  methods  used  for  the  determination  of  latent  heat  are  essen- 
tially similar  to  those  used  for  the  determination  of  specific  heat. 


CHANGE    OF    TEMPERATURE    AND    CHANGE    OF    STATE.  333 

De  la  Provostaye  and  Desains,  and  Regnault,  found  the  latent 
heat  of  fusion  of  ice  to  be  equal  to  79*25  units.  Person,  more  re- 
cently, has  used  a  different  method,  ultimately  with  the  same  result. 
He  heated  a  quantity  of  ice,  the  temperature  of  which  was  originally 
below  0°  C. ;  and,  in  consequence,  he  had  to  take  account  of  the 
specific  heat  of  ice,  which,  as  we  have  seen,  is  about  0"5°.  He  at 
first  obtained  the  value  80  for  the  amount  of  latent  heat  of  fusion ; 
but,  subsequently,  he  traced  the  discrepancy  between  his  result  and 
that  of  previous  observers  to  the  fact  that  latent  heat  seems  to  be 
absorbed  to  a  slight  extent)  before  the  temperature  0°  is  reached. 
This,  if  true,  furnishes  evidence  of  the  truth  of  the  supposition, 
made  in  §  78,  that  the  process  of  liquefaction  is  gradual. 

The  following  values  of  the  latent  heat  of  fusion  of  some  sub- 
stances are  taken  from  Person's  results  : 

Latent  Heat  of  Fusion. 

Ice     79-25         Tin  14-25 

Phosphate  of  soda  ...     66-80        Lead        5'37 

Zinc 28-13         Mercury 2'83 

275.  Evaporation  and  Condensation. — The  laws  of  evaporation 
are  similar  to  those  of  fusion. 

1.  So  long  as  the  pressure  is  maintained  constant,  there  is  a 
definite  boiling-point  for  every  liquid. 

2.  If  the  liquid  be  well  stirred,  the  temperature  of  both  liquid 
and  vapour  remains  at  the  boiling-point  until  all  the  liquid  has 
evaporated. 

The  effect  of  pressure  is  always  in  one  direction  with  regard  to 
the  boiling-point,  for  all  substances  expand  when  they  evaporate. 
The  effect  is  therefore  to  raise  the  boiling-point,  and  its  elevation  is 
much  more  marked  than  is  the  alteration  of  the  melting-point  of  a 
substance.  But,  before  discussing  this  point  farther,  we  must  con- 
sider more  fully  the  process  which  is  termed  boiling. 

Evaporation  occurs  to  a  greater  or  less  extent  at  all  temperatures, 
and  the  rate  of  evaporation,  ceteris  paribus,  increases  rapidly  as  the 
temperature  rises.  If  the  liquid  be  contained  in  a  closed  vessel,  the 
rate  of  evaporation  gradually  decreases  and  finally  vanishes.  (We 
assume,  of  course,  that  the  area  from  which  evaporation  takes  place 
remains  constant.  Under  given  conditions',  the  total  rate  of  evapora- 
tion is  proportional  to  the  magnitude  of  the  area.)  It  is  not  really 
true  that  evaporation  has  ceased.  A  state  of  kinetic  equilibrium,  in 
which  the  rate  of  evaporation  is  equal  to  the  rate  of  condensation, 
has  been  attained.  When  this  condition  of  equilibrium  holds,  the 


334 


A    MANUAL    OF    PHYSICS. 


vapour  is  said  to  be  saturated ;  and  it  is  found  that  the  pressure  of 
the  saturated  vapour  depends  only  on  the  temperature. 

The  presence  of  gases,  such  as  air,  has  no  influence  upon  the 
final  state  of  equilibrium  :  it  merely  increases  the  time  necessary 
for  the  attainment  of  the  condition. 

But  the  vapour  may  be  saturated  at  any  temperature  as  well  as 
at  the  usual  boiling-point.  And  this  leads  to  the  definition  of  the 
boiling-point  as  the  temperature  at  which  the  pressure  of  the 
saturated  vapour  is  equal  to  that  to  which  the  free  surface  of  the 
liquid  is  subjected. 

The  following  remarks  should  make  the  matter  clear.  Let 
ABCD  (Fig.  158)  represent  a  cylinder  in  which  a  smooth,  massless 
(and  therefore  weightless)  piston  AD,  which  we  also  suppose  to 
be  gas-tight,  works  freely.  First,  let  there  be  a  gas  in  the 
closed  region  P  and  another  gas  in  the  region  Q  outside  the 


Q 


FIG.  158. 

piston.  Evidently  equilibrium  is  only  reached  when  the  pressure  is 
the  same  on  both  sides  of  the  piston.  Now  suppose  that  P  is  filled 
with  a  liquid  below  its  boiling-point.  The  filling  of  the  region  P  is 
necessarily  complete  so  long  as  vapour  is  not  formed.  And  no 
vapour  can  be  formed  until  the  pressure  of  that  vapour  is  equal  to 
the  pressure  of  the  gas  in  the  region  Q,  i.e.,  until  its  pressure  is 
equal  to  the  pressure  to  which  the  free  surface  of  the  liquid  is  ex- 
posed. But  when  the  liquid  is  at  the  temperature  at  which  this 
occurs,  vapour  will  be  formed,  and  the  continued  application  of 
heat  will  force  the  piston  up.  If  we  now  suddenly  produce  a 
vacuum  in  the  region  Q,  vapour  will  be  rapidly  formed  in  P ;  and 
that  vapour  will  proceed  not  merely  from  the  surface  of  the  liquid 
but  also  in  bubbles  from  its  interior.  This  process  of  free  evapora- 
tion is  called  boiling  or  ebullition. 

[The  phenomena  exhibited  by  Geysers  are  due  to  a  like  cause.     A 
sudden  reduction  of  pressure  in  the  interior  of  the  column  of  water 


CHANGE    OF    TEMPERATURE    AND    CHANGE    OF    STATE.  335 

which  fills  the  funnel  causes  the  water  at  that  part  to  change  its 
state  explosively,  and  so  the  superincumbent  water"  is  ejected 
violently.] 

The  following  table,  given  by  Begnault,  exhibits  the  relation 
between  the  boiling-point  and  the  pressure  : 

Pressure  of  the  Saturated  Vapour  of  Water. 

Temp.  C.                       Pressure  in  Temp.  C.                         Pressure  in 
Atmospheres.                                          Atmospheres. 

0°  0-006  120°  1*962 

10°  0-012  130°  2-671 

20°  0-023  140°  3-576 

30°  0-042  150°  4-712 

40°  0-072  160°  6-120 

50°  0-121  170°  7-844 

60°  0-196  180°  ...         ...       9-929 

70°  0-306  190°  12-425 

80°  0-466  200°  15-380 

90° 0-691  .  210°  18-848 

100°  1-000  220°  22-882 

110°  1-415  230°  27-535 

By  sufficiently  reducing  the  pressure,  water  may  be  made  to  boil 
violently — not  merely  to  evaporate  at  its  surface — at  temperatures 
far  below  its  ordinary  boiling-point.  The  well-known  experiment 
of  causing  hot  water  to  boil  in  a  closed  flask,  by  pouring  cold 
water  upon  the  flask,  is  a  case  in  point.  The  sudden  reduction 
of  temperature  causes  partial  condensation  of  the  vapour  already 
formed  in  the  flask,  and  so  gives  rise  to  a  sudden  diminution  of 
pressure. 

When  the  boiling-point  is  known,  the  atmospheric  pressure  may 
be  obtained  from  a  table  such  as  that  above.  The  Hypsometric 
Thermometer,  used  for  the  determination  of  height  above  sea -level, 
is  based  upon  this  principle.  The  atmospheric  pressure  diminishes 
as  the  elevation  above  sea-level  increases,  and  the  result  is  that  the 
boiling-point  is  lowered  by  about  1°  C.  at  an  elevation  of  960  feet 
above  sea-level. 

The  laws  of  evaporation  are  subject  to  exceptions,  just  as  are  the 
laws  of  melting.  Thus  water  may,  by  cautious  heating  in  a  smooth 
clean  glass  vessel,  be  raised  considerably  above  its  ordinary  boiling- 
point,  if  it  has  been  carefully  freed  from  dissolved  gases.  A  very 
slight  vibration  may  then  cause  it  to  boil  explosively. 


336 


A   MANUAL    OF    PHYSICS. 


The  boiling-points  of  various  liquids  differ  greatly  under  ordinary 
atmospheric  conditions,  as  the  following  table  shows  : 

Table  of  Boiling-points  of  Liquids. 

Zinc     1040°  C.         Bisulphide  of  carbon     ...       48°  C. 

Mercury      350°  Sulphurous   acid    -10° 

Water 100°  Nitric  oxide  -87° 

276.  Latent  Heat  of  Vaporisation.  —  Eegnault  found  for  the 
'  total  heat  of  steam,'  i.e.,  the  quantity  of  heat  which  is  given  out 
by  one  pound  of  water  in  condensing  to  water  at  0°  C.,  the  expres- 
sion, 

H  =  606-5  -f  0-305£, 

where  t  is  the  temperature  in  Centigrade  degrees.     This  gives  for 
the  latent  heat  the  expression 


•  H-f'adt, 


where  a  is  the  specific  of  water.  The  value  of  a  is  (§  269) 
1 + 0'00004£  +  0'0000009£2.  If  we  substitute  this  value  in  the  integral, 
we  get 

L  =  606-5  -  0'695£  -  Q-00002*2  -0-0000003^. 

This  formula  is  true  throughout  the  range  of  temperature  from 
0°  C.  to  230°  C.  If  we  could  assume  that  it  held  true  up  to  706°  C., 
it  would  indicate  that  the  latent  heat  vanishes  at  that  temperature 
very  nearly.  (See  §  278.) 

The  latent  heat  of  steam  is  very  large  in  comparison  with  that  of 
most  other  liquids,  as  this  table  shows  : 

Latent  Heat  of  Valorisation. 

Water 536        Ether 90;4 

Naphtha  264         Bisulphide  of  carbon        86*7 

Alcohol 202        Bromine          45-6 

We  have  found  previously  that  the  latent  heat  of  liquefaction  of  ice 
is  also  relatively  large.  These  facts — of  the  large  latent  heats  of 
liquefaction  of  ice  and  of  vaporisation  of  water — are  of  great  im- 
portance in  the  economy  of  nature.  If  they  were  not  so,  destruc- 
tive floods  might  frequently  occur  from  the  rapid  liquefaction  of  ice, 
or  sudden  condensation  of  moisture,  consequent  on  a  slight  variation 
of  temperature. 

The  latent  heat  of  vaporisation  is  used  for  the  production  or 
maintenance  of  low  temperatures.     Water  may  be  kept  cool  in  very 


CHANGE    OF    TEMPERATURE    AND    CHANGE    OF    STATE.  337 

hot  weather  if  it  is  enclosed  in  a  vessel  of  porous  earthenware  ;  for 
part  of  it  percolates  through  the  vessel  and  evaporates  from  its 
outer  surface,  the  latent  heat  being  largely  drawn  from  the  vessel 
and  its  liquid  contents.  Also  solid  carbonic  acid  is  produced  if  a 
jet  of  the  liquid  (formed  under  considerable  pressure)  is  allowed 
to  escape  from  the  vessel  which  contains  it ;  for  the  outer  parts  of 
the  jet  evaporate,  and  the  necessary  latent  heat  is  largely  taken 
from  the  interior  parts  of  the  jet,  which  consequently  are  solidified. 
Faraday  froze  mercury  in  the  interior  of  a  white-hot  platinum 
crucible,  by  placing  it  in  a  capsule  which  rested  on  a  mixture  of 
solid  carbonic  acid  and  ether  contained  in  the  crucible.  Similarly, 
in  very  hot  countries,  ice  may  be  formed  at  night  on  shallow  pools 
because  of  rapid  evaporation. 

277.  Formation  of  Dew. — When  a  superheated  vapour  is  cooled 
sufficiently,  saturation  takes  place,  and  any  further  cooling  causes 
condensation.  The  moisture  which  is  deposited  in  this  way  from 
the  atmosphere  is  termed  dew.  Any  cold  body  lowers  the  tempera- 
ture of  the  air  in  immediate  contact  with  it ;  and,  when  the  tempera- 
ture is  sufficiently  lowered,  a  thin  film  of  moisture  is  deposited  upon 
the  cold  body.  The  latent  heat  which  is  given  out  on  condensation 
gradually  raises  the  temperature  of  the  cold  body  until  it  becomes 
equal  to  that  which  corresponds  to  the  vapour-pressure,  at  which 
stage  the  action  ceases.  This  temperature,  being  also  that  at  which 
the  deposition  of  dew  just  commences,  is  called  the  Dew-point. 
Hoar-frost  is  formed  when  the  dew-point  is  below  0°  C. 

Wells  first  gave  the  correct  explanation  of  the  formation  of  dew. 
He  showed  that  dew  is  freely  deposited  on  nights  when  the  sky  is 
clear,  because  on  such  nights  the  earth  loses  heat  rapidly  by  radia- 
tion and  so  cools  rapidly  to  the  dew-point ;  whereas,  on  cloudy 
nights,  the  clouds  absorb,  and  radiate  back  to  the  earth,  a  large 
part  of  the  radiated  heat,  so  that  the  ground  does'  not  cool  rapidly. 
Another  condition  necessary  to  the  ready  formation  of  dew  is  that 
the  air  shall  be  still,  otherwise  no  portion  of  the  air  may  remain  in 
contact  with  the  ground  for  a  length  of  time  sufficient  to  allow  of 
its  being  cooled  to  the  dew-point.  The  dew  will,  of  course,  deposit 
itself  most  freely  on  those  bodies  which  part  with  their  heat  most 
rapidly  and  have  also  small  specific  heat. 

Aitken  has  recently  shown  that  the  presence  of  particles  of  dust  is 
necessary  before  condensation  of  moisture  can  occur  in  the  atmo- 
sphere, and  that  supersaturation  of  a  vapour  can  be  produced  by 
getting  rid  of  all  dust-particles  by  filtration  through  cotton-wool. 
These  particles  act  as  nuclei  upon  which  the  deposition  takes  place. 
This  phenomenon  is  very  closely  connected  with  the  phenomenon  of 

22 


338  A    MANUAL    OF    PHYSICS. 

the  dependence  of  the  equilibrium-pressure  of  vapour  upon  the  cur- 
vature of  the  liquid  film  with  which  it  is  in  contact  (§  127). 

Aitken  has  also  utilised  the  fact  that  moisture  is  deposited  upon 
the  dust-particles  in  the  construction  of  an  instrument  which 
enables  us  to  determine  the  number  of  particles  which  are  contained 
in  a  given  volume  of  any  definite  specimen  of  air.  This  instrument 
is  certain  to  prove  of  considerable  meteorological  importance. 

Daniell's  Hygrometer  was  constructed  for  the  purpose  of 
accurately  registering  the  dew-point.  It  consists  of  two  hollow 
glass  bulbs  connected  by  a  glass  tube.  One  of  these  bulbs  is  made  of 
black  glass,  and  the  other  is  made  of  clear  glass.  A  small 
thermometer,  the  stem  of  which  projects  into  the  tube  of  (clear) 
glass  which  connects  the  two  bulbs,  is  placed  in  the  black  bulb 
along  with  a  quantity  of  sulphuric  ether.  The  remaining  portions 
of  the  interior  of  the  instrument  are  filled  only  with  the  vapour  of 
ether.  A  piece  of  cambric  is  tied  round  the  other  bulb,  and  a  little 
ether  is  poured  upon  it.  The  evaporation  of  this  ether  cools  the 
bulb,  and  makes  some  of  the  vapour  inside  it  condense.  This  de- 
stroys the  equilibrium  of  the  liquid  ether  and  its  vapour,  in  the 
interior  of  the  instrument ;  and  some  of  the  ether  in  the  black  bulb 
evaporates,  in  order  'that  equilibrium  may  be  restored.  The  absorp- 
tion of  latent  heat  cools  this  bulb,  and,  finally,  dew  is  deposited  on 
its  exterior.  The  presence  of  a  very  slight  film  of  dew  is  readily 
observed  on  the  black  surface,  and  the  temperature  of  the  bulb  is 
noted  ;  but  the  reading  of  the  thermometer  is  necessarily  a  little  too 
low.  The  evaporation  is  then  stopped,  and  the  temperature  at 
which  the  dew  just  disappears  is  observed.  This  reading  is  a  little 
too  high,  and  so  the  mean  of  the  two  results  is  taken. 

Kegnault  introduced  improvements  which  rendered  it  possible  to 
observe  the  appearance  and  disappearance  of  the  dew  at  practically 
one  temperature. 

The  dew-point  is  also  found  by  means  of  Wet  and  Dry  Bulb 
Thermometers.  The  one  thermometer  has  its  bulb  surrounded  by 
cambric,  which  is  kept  moist  with  water  drawn  up  by  capillary  action 
through  some  threads  which  dip  into  a  vessel  containing  it.  The 
other  (ordinary)  thermometer  registers  the  exact  temperature  of  the 
air.  The  reading  of  the  wet-bulb  thermometer  is  lower  than  that  of 
the  dry -bulb  thermometer  so  long  as  evaporation  is  going  on ;  but,  if 
the  atmosphere  is  saturated  with  water-vapour,  no  evaporation  takes 
place,  and  both  thermometers  register  the  same  temperature.  The 
formula 

S       b 
^=^-48 '30' 


CHANGE    OF    TEMPERATURE    AND    CHANGE    OF    STATE.  339 

in  which  p  represents  the  pressure  of  water-vapour  in  the  atmosphere, 
pa  represents  the  pressure  which  is  given  in  Regnault's  table  as  cor- 
responding to  the  temperature  of  the  wet-bulb,.  S  is  the  difference 
between  the  wet-bulb  and  the  dry-bulb  readings,  and  b  is  the  height 
of  the  barometric  column  in  inches,  was  found  by  Apjohn  to  accord 
well'  with  observed  results. 

278.  Continuity  of  the  Liquid  and  Gaseous  States.  Critical 
Temperature. — Cagniard  de  la  Tour  first  showed  that  a  substance 
may  exist  in  a  non-liquid  state  at  a  density  very  nearly  equal  to  its 
density  in  the  liquid  condition.  A  complete  investigation  of  the 
subject  was  made  by  Andrews,  who  showed  that 

There  is  a  Critical  Temperature  for  every  vaporous  or 
gaseous  substance,  such  that  no  amount  of  pressure  can  liquefy  the 
substance,  unless  its  temperature  be  below  the  critical  value. 

The  critical  temperature  of  carbonic  acid  is  30°*9  C.  That  of 
water  is  about  412°  C. 

The  latent  heat  vanishes  at  the  critical  temperature.  We  have 
already  seen  that  the  latent  heat  of  water  should  vanish  at  about 
706°  C.,  if  Regnault's  formula  connecting  latent  heat  with  tempera- 
ture held  throughout  that  range.  The  result  just  given  shows  that 
the  formula  deviates  largely  from  the  truth  at  temperatures  higher 
than  the  limit  (230°  C.)  up  to  which  Regnault  worked. 

The  accompanying  diagram  (Fig.  159)  represents  the  results  of 
Andrews'  experiments  on  carbonic  acid.  Pressure  is  measured  (in 
atmospheres)  along  the  axis  of  ordinates,  and  volume  is  measured 
along  that  of  abscissae.  At  the  temperature  13°'1,C.,  the  volume  of 
the  gas  gradually  diminishes,  as  the  pressure  is  raised,  until  lique- 
faction commences.  After  this,  the  volume  lessens,  without  any 
rise  of  pressure,  until  all  the  substance  is  liquefied;  and  then 
immense  pressure  is  required  to  lessen  it  even  slightly. 

Similar  effects  take  place  at  the  higher  temperature  21°'5.  The 
line  (called  an  isothermal)  which  represents  the  simultaneous  values 
of  pressure  and  volume  at  this  higher  temperature,  lies,  in  the 
diagram,  entirely  to  the  right  hand  of,  and  above,  the  isothermal  of 
13°'l ;  for,  the  pressure  being  constant,  the  volume  increases  with 
the  temperature,  and,  the  volume  being  constant,  the  pressure 
increases  with  the  temperature.  But,  at  this  higher  temperature, 
the  change  of  volume  in  passing  from  the  gaseous  to  the  liquid 
state  is  smaller  than  that  which  occurs  at  the  lower  temperature  \ 
and  liquefaction  commences  at  a  smaller  volume,  and  ends  at  a 
larger  volume,  than  when  the  temperature  is  less.  The  isothermals 
cease  to  have  a  portion  parallel  to  the  axis  of  volume,  i.e.,  liquefac- 
tion ceases,  at  30°'9. 

22—2 


340  A   MANUAL    OF    PHYSICS. 

The  dotted  curve  separates  the  region  in  which  the  liquid  and  the 
vapour  can  exist  together  in  equilibrium  from  the  regions  in  which 
the  substance  is  entirely  liquid  or  entirely  vapour.  The  isothermal 
of  .300>9  separates  the  region  in  which  liquefaction  can  occur  from 
that  in  which  it  is  impossible. 

[Compare,  with  this  diagram,  Fig.  11,  which  is  drawn,  so  as 
approximately  to  suit  the  case  of  carbonic  acid,  from  theoretical 
considerations  based,  by  Professor  Tait,  upon  the  kinetic  theory  of 
gases.] 

We  may  with  great  advantage,  as  Tait  suggests,  describe  the 
substance  as  a  true  gas,  or  a  true  vapour,  according  as  the  tempera- 
ture is  higher,  or  lower,  than  the  critical  temperature. 

The  compressibility  of  the  substance  is  dv/vdp,  where  v  is  the  volume 
and  dv,  dp,  represent  respectively  simultaneous  small  increments  of 
the  volume  and  the  pressure.  Now,  the  diagram  shows  that,  at  the 
commencement  of  liquefaction,  the  inclination  of  the  isothermal  to 
the  axis  of  volume  becomes  greater  and  greater  as  the  temperature 
rises,  i.e.,  the  ratio  dv/dp  decreases  as  the  temperature  rises.  Hence, 
since  we  suppose  unit  volume  to  be  taken  in  all  cases,  the  com- 
pressibility of  the  vapour  when  it  is  upon  the  point  of  condensing 
decreases  as  the  critical  temperature  is  approached.  Similarly,  the 
value  of  dvjdp  in  the  liquid  state,  when  the  substance  has  just  been 
entirely  liquefied,  increases  as  the  temperature  rises ;  and  thus  we 
see  that  the  compressibility  in  the  two  states  tends  towards  equality, 
simultaneously  with  the  volumes,  as  the  temperature  rises  to  its 
critical  value. 

For  a  considerable  distance  above  the  critical  point  the  isother- 
mals  exhibit  two  points  of  inflexion ;  but  these  finally  cease  to  be 
visible,  and  the  isothermals  closely  resemble  those  of  a  perfect  gas. 

[The  illustration  affords  a  good  example  of  the  use  of  contours. 
The  isothermals  may,  as  was  stated  in  Chap.  III.,  be  regarded  as 
the  projections  of  the  plane  sections  of  a  surface  which  represents 
the  various  simultaneous  values  of  the  pressure,  volume,  and  tem- 
perature of  the  gas.] 

279.  Solution.  Freezing  Mixtures. — The  process  of  solution  is 
extremely  analogous  to  the  processes  of  liquefaction.  A  gas  which 
is  dissolved  in  a  liquid  may  be  regarded  to  a  certain  extent  as  if  it 
were  liquefied,  and  latent  heat  is  given  out  in  the  process  of  solution. 
Similarly,  when  a  solid  is  dissolved  in  a  liquid,  latent  heat  is  required, 
just  as  if  the  solid  were  directly  liquefied  ;  but  in  some  cases  this 
absorption  of  latent  heat  is  masked  by  the  heat  which  is  developed 
because  of  molecular  action  between  the  liquid  and  the  solid. 

The  amount  of  heat  which  is  disengaged  in  the  solution  of  a  gas 


CHANGE    OF   TEMPERATURE   AND    CHANGE   OF   STATE. 


841 


342  A   MANUAL   OF   PHYSICS. 

is  frequently  very  marked.  This  is  so  specially  in  the  cases  of  the 
more  soluble  gases,  such  as  ammonia  when  water  is  the  solvent. 

The  amount  of  gas,  under  definite  pressure,  which  a  given  liquid 
will  dissolve,  becomes  less  as  the  temperature  is  raised  ;  though,  by 
careful  treatment,  a  state  of  supersaturation  may  be  induced — which 
is  analogous  to  the  prevention  of  boiling  at  temperatures  consider- 
ably over  the  ordinary  boiling-point  of  a  liquid. 

Supersaturation  of  a  liquid  solution  of  a  solid  may  also  take  place 
— notably  in  the  case  of  a  substance,  such  as  acetate  of  soda,  which 
dissolves  in  little  more  than  its  own  water  of  crystallisation.  If  a 
crystal  of  the  acetate  be  dropped  into  the  supersaturated  solution  to 
act  as  a  nucleus,  crystallisation  will  take  place  rapidly  with  the 
development  of  latent  heat.  A  crystal  of  any  other  substance  of 
the  same  crystalline  form  will  produce  the  same  effect. 

Heat  is  frequently  developed  or  absorbed  when  two  liquids  are 
mixed  (mutually  dissolved).  If  chemical  action  takes  place  to  any 
extent  between  the  two,  heat  will  be  developed  unless  other  causes 
prevent.  If  the  total  bulk  of  the  two  liquids  increases  on  mixture — 
as  in  the  case  of  bisulphide  of  carbon  and  alcohol — heat  tends  to  be 
absorbed ;  and  again,  the  water-equivalent  of  the  mixture  may  be 
greater  than  the  sum  of  the  water-equivalents  of  its  constituents — 
which  also  necessitates  absorption  of  heat.  If  the  opposite  effects 
to  these  take  place,  heat  will  be  evolved.  Energy  may  also  be 
changed  into  heat  in  the  process  of  inter-diffusion  of  the  liquids. 
Disengagement  or  absorption  of  heat  take  place  respectively  accord- 
ing as  the  effects  of  the  one  or  the  other  sets  of  actions  preponderate. 
And,  as  the  various  actions  depend  upon  the  temperature,  we  find 
that  the  total  effect  is  sometimes  reversed  when  the  original  tem- 
perature of  the  two  liquids  is  sufficiently  varied. 

Two  solids  even  may  dissolve  in  each  other,  so  to  speak,  with  the 
absorption  of  latent  heat.  (Salt  and  snow  furnish  a  well-known 
example.)  This  can  (Jbviously  only  occur  when  the  freezing-point 
of  the  resultant  liquid  fe  lower  than  the  original  (common)  tempera- 
ture of  the  solids.  Part  of  the  latent  heat  is  obtained  by  cooling 
the  solids,  part  by  cooling  the  liquid,  and  part,  it  may  be,  by  cooling 
surrounding  bodies.  If  the  whole  be  intimately  mixed  the  action 
necessarily  ceases  when  the  freezing-point  of  the  resultant  liquid  is 
reached.  These  remarks  contain  the  explanation  of  the  action  of 
solid  freezing-mixtures,  which  has  been  elaborately  investigated  by 
Professor  Frederick  Guthrie. 

280.  Dissociation  and  Chemical  Combination. — When  the  tem- 
perature is  raised  sufficiently  high  a  compound  dissociates,  or  breaks 
up,  into  its  constituents.  The  change  is  not  sudden  but  gradual. 


CHANGE     OF    TEMPERATURE    AND    CHANGE    OF    STATE.  843 

It  commences  at  a  certain  lower  limit  of  temperature,  and  ends 
completely  at  a  certain  higher  limit ;  and  at  all  intermediate  tem- 
peratures a  state  of  kinetic  equilibrium  is  arrived  at  in  which  recom- 
bination precisely  balances  dissociation.  The  magnitude  of  the 
limits  will,  in  general,  depend  upon  the  pressure.  It  is  usual  to 
speak  of  the  temperature  at  which  one-half  of  the  substance  is  dis- 
sociated as  the  temperature  of  dissociation. 

Conversely,  when  the  two  (or  more)  constituents  are  mixed,  com- 
bination does  not  occur  until  a  certain  temperature  is  attained ;  but, 
if  the  combination  results  in  the  development  of  heat,  the  process, 
once  started,  will  continue  until  the  percentage  of  the  mixture  which 
remains  uncombined  corresponds  to  the  temperature  which  the 
whole  mass  attains  because  of  the  heat  which  is  set  free.  On  the 
other  hand,  if  work  is  done  during  the  process,  or  if  heat  is  lost  by 
conduction  or  otherwise,  the  process  will  continue  until  combination 
is  complete  when  the  temperature  falls  to  the  lower  limit. 

All  chemical  combination  takes  place  in  accordance  with  the  two 
laws  of  thermodynamics,  and  therefore  further  treatment  of  this 
subject  is  deferred  until  we  have  considered  these  laws.  (See 
§  298.) 

281.  Many  other  effects  of  heat  might  be  noted  here,  but  it  is 
preferable  to  leave  their  discussion  to  those  special  sections  in  which 
we  have  to  treat  of  the  properties  which  are  affected  by  the  applica- 
tion of  heat. 


CHAPTEE  XXIV. 

CONDUCTION   AND    CONVECTION    OF    HEAT. 

282.  Conduction. — We  have  already  discussed  the  transference  of 
heat  by  the  process  of  radiation,  that  is,  the  transference  of  heat 
without  the  mediation  of  ordinary  matter.  In  the  process  of  radia- 
tion the  transferred  energy  may  pass  through  a  material  substance 
without  being  communicated  to  it ;  indeed,  radiation  ceases  in  so 
far  as  such  communication  is  made.  We  must  now  consider  its 
transference  when  ordinary  matter  is  the  medium  through  which  it 
is  transferred. 

The  most  marked  difference  between  the  two  cases  lies  in  the  rate 
of  propagation,  which  is  extremely  rapid  when  radiation  occurs, 
while  it  is  extremely  slow  in  comparison  when  ordinary  matter  is 
the  medium  of  transference. 

Two  methods  exist  according  to  which  heat  (which  consists  in 
kinetic  energy  of  molecular  motion)  may  pass  from  one  place  to 
another  by  means  of  matter.  The  energy  may  pass  from  one  por- 
tion of  matter  to  another,  which  occupies  a  different  position,  by 
actual  (or  virtual)  impact  between  the  molecules  of  the  two  portions  ; 
that  is  to  say,  it  may  be  handed  on  from  one  portion  to  another  :  or 
again,  it  may  pass,  not  from  one  portion  of  matter  to  another,  but  from 
one  locality  to  another  by  motion  of  the  hot  body.  The  former  of 
these  processes  is  known  as  Conduction ;  the  latter  as  Convection. 
Both  take  place  in  liquids  and  in  gases  ;  the  former  alone  can  take 
place  in  solids. 

283.  Conductivity. — Different  substances,  under  like  conditions, 
conduct  heat  at  different  rates.  Thus  a  bar  of  iron,  one  end  of 
which  is  red-hot,  may  be  too  hot  to  grasp  at  the  cooler  end  ;  while 
a  bar  of  wood,  of  the  same  length,  which  is  burning  at  one  end,  may 
be  easily  handled  at  the  other.  The  property  in  virtue  of  which 
such  differences  arise  is  termed  Conductivity. 

Most  experiments  which  are  intended  to  illustrate  the  differences 
between  the  conductivities,  or  conducting-powers,  of  various  sub- 


CONDUCTION    AND    CONVECTION    OF    HEAT.  345 

stances  for  heat  exhibit  only  the  differences  between  the  rates 
at  which  the  temperatures  of  the  substances,  at  a  given  distance 
from  the  source  of  heat,  attains  a  definite  value  under  given 
conditions.  The  well-known  experiment  of  Ingenhouz  is  of  this 
description.  In  it,  a  series  of  similar  and  equal  rods,  of  different 
substances,  project  from  the  side  of  a  metallic  trough  into  which  hot 
water  is  suddenly  poured.  Each  rod  is  coated  with  a  thin  film  of 
beeswax  which  melts  at  a  definite  temperature.  The  rate  at  which 
this  definite  temperature  travels  along  each  rod  is  plainly  shown  by 
the  motion  of  the  line  of  demarcation  between  the  melted  and  the 
unmelted  portions  of  the  wax.  But  obviously  this  rate  will  only 
coincide  with  the  rate  at  which  heat  is  conducted  along  when  the 
thermal  capacities  of  the  various  substances  are  practically  identical, 
for,  other  things  being  equal,  the  rate  at  which  the  temperature  rises 
is  inversely  proportional  to  the  thermal  capacity. 

Fourier  was  the  first  to  give  an  accurate  definition  of  conductivity. 
The  whole  subject  of  heat-conduction  was  so  fully  and  accurately 
developed  by  him  that  his  work  —  '  Theorie  analytique  de  la  Chaleur  ' 
—  published  in  1822,  still  remains  the  text-book  on  the  subject. 

Let  us  suppose  that  the  substance,  the  thermal  conductivity  of 
which  we  are  considering,  is  in  the  form  of  a  uniformly  thick  plane 
slab  of  practically  infinite  extent.  Let  9  be  its  thickness  ;  and  let 
one  side  be  kept  at  uniform  temperature  t,  while  the  other  is  kept 
at  uniform  temperature  t'  until  a  steady  flow  of  heat  takes  place 
from  side  to  side.  The  quantity  of  heat,  h,  which  passes  in  T  units 
of  time  through  an  area  a  of  the  surface  of  the  slab  is  found  ex- 
perimentally to  be  directly  proportional  to  r,  a,  and  t'  —  t,  while  it  is 
inversely  proportional  to  9.  Hence  we  may  write 


The  quantity  (£'  —  £)/#  is  called  the  temperature  gradient,  and  Jc  is 
the  conductivity. 

If  the  area,  the  temperature  gradient,  and  the  time,  be  all  unity, 
the  equation  becomes 

&»*, 
and  so  we  obtain  the  following  definition  of  the  conductivity  : 

The  thermal  conductivity  of  a  substance,  at  any  temperature,  is 
the  number  of  units  of  heat  which  pass,  per  unit  of  time,  through 
unit  of  surface  of  an  infinite  slab  of  the  substance,  of  unit  thick- 
ness, the  sides  of  which  are  kept  respectively  at  temperatures  half 
a  degree  higher,  and  half  a  degree  lower,  than  the  given  tempera- 
ture. 


346  A    MANUAL    OF    PHYSICS. 

In  making  this  definition  we  assume  that  the  unit  of  length  is  not 
excessively  small — that  it  is  (say)  a  centimetre,  an  inch,  or  a  foot — 
and  that  an  ordinary  temperature  degree — say,  the  Centigrade — is 
used,  so  that  the  temperature  gradient  is  not  large.  The  necessity 
for  these  restrictions  is  apparent,  if  we  consider  that  the  conductivity 
may  (it  actually  does)  vary  somewhat  with  the  temperature  ;  for,  in 
consequence  of  such  variation,  the  temperature  gradient  could  not 
be  sensibly  uniform  from  side  to  side  of  the  slab,  if  the  difference  of 
the  temperatures  at  the  two  sides  were  large.  As  a  special  case, 
let  us  suppose  that  the  conductivity  of  a  layer  of  the  slab,  of  half  its 
total  thickness,  is  one-half  of  that  of  the  remaining  portion.  Since 
the  same  flow  of  heat  takes  place  through  both  portions,  the  dif- 
ference of  temperature  between  the  sides  of  the  former  portion  must 
be  double  of  that  between  the  sides  of  the  latter. 

Of  course,  even  if  the  conductivity  varies  from  point  to  point, 
whether  from  variation  of  temperature  or  from  any  other  cause,  the 
quantity  &,  determined  from  the  above  formula,  will  always  repre- 
sent the  average  conductivity  of  the  slab  considered  as  a  whole. 

But,  quite  apart  from  the  question  of  such  variation,  we  cannot 
assert  that  the  quantity  of  heat  which  will  pass  through  a  slab,  one 
unit  in  thickness,  under  unit  difference  of  temperature,  will  be  pre- 
cisely equal  to  the  quantity  which  will  pass  through  a  slab,  n  times 
thinner,  under  a  difference  of  temperature  n  times  less,  when  n  is 
a  very  large  number,  and  all  the  other  conditions  are  unaltered. 

284.  Measurement  of  Conductivity. — In  one  form  of  experiment 
for  the  absolute  determination  of  conductivity,  a  steady  state  of 
temperature  is  maintained  throughout  the  substance.  This  method 
was  used  by  Lambert,  and  subsequently,  under  greatly  improved 
conditions,  by  Forbes. 

In  Forbes'  method  a  long  bar  of  the  substance  of  uniform  cross- 
sectional  area  is  used.  One  extremity  of  the  bar  is  inserted  in  a  bath 
of  melted  lead,  or  solder ;  and  the  other  extremity  is  exposed  to  the 
air,  or,  if  necessary,  is  cooled  by  a  current  of  water.  Small  holes, 
into  which  a  little  mercury  is  poured,  are  drilled  in  the  bar  at 
regular  intervals;  and  these  holes  are  lined  with  iron  (if  the  bar 
itself  be  not  made  of  iron)  in  order  to  prevent  amalgamation. 
Thermometers,  inserted  in  the  holes  (which  are  found  not  to  appre- 
ciably affect  the  flow  of  heat  along  the  bar)  register  the  temperature 
of  the  bar  in  their  immediate  vicinity. 

If  distance  measured  along  the  bar  from  the  source  of  heat  be  laid 
off  along  ox  (Fig.  160),  and  if  ordinates  be  drawn  at  points  such  as 
j>,  corresponding  to  the  positions  of  the  thermometers,  and  of  lengths 
which  are  proportional  to  the  readings  of  the  thermometers  at  these 


CONDUCTION   AND    CONVECTION    OF   HEAT.  347 

points,  a   curve  drawn  free-hand  through  the    extremities   of   the 
ordinates  will  enable  us  to  obtain  the  temperature  gradient  at  any 


P 
FIG.  160. 

part  of  the  bar.  For  the  tangent  of  the  angle  of  inclination  of  the 
line  which  touches  the  curve  at  the  extremity  of  any  ordinate  is 
equal  to  the  space  rate  at  which  the  temperature  varies,  per  unit  of 
length,  at  the  corresponding  section  of  the  bar  ;  i.e.,  it  is  equal  to 
the  gradient  of  temperature  at  that  section.  But  the  sectional  area 
of  the  bar  is  known,  and  hence,  if  we  can  determine  the  quantity  of 
heat  which  passes  in  a  given  time  through  the  given  section,  we 
can  determine  the  conductivity  by  calculation  from  the  equation 
above. 

Now  the  heat  which  passes  any  section  is  entirely  lost  from  the 
remaining  portion  of  the  bar  by  radiation,  or  otherwise ;  and  any 
heat  which  is  given  to  the  water  employed  in  cooling  the  far  end  of 
the  bar,  if  this  is  required,  can  be  readily  estimated  by  means  of  the 
rise  in  temperature  of  the  water,  while  the  heat  which  is  lost  by 
radiation  and  convection  is  found  by  a  special  experiment. 

During  the  above  experiment,  a  thermometer,  inserted  in  a  hole 
in  a  small  bar  which  is  cut  originally  from  the  long  bar,  indicates 
.the  temperature  of  the  air  in  the  neighbourhood  of  the  large  bar. 
In  the  second  experiment,  which  is  made  for  the  purpose  of  deter- 
mining the  rate  of  loss  of  heat,  the  small  bar  is  heated  uniformly  to 
a  temperature  higher  than  the  highest  recorded  in  the  former  one. 
The  bar  is  now  allowed  to  cool,  and  the  thermometer  which  is 
inserted  in  it  enables  us  to  determine  the  rate  of  loss  of  heat  per 
unit  of  time,  per  unit  of  length  of  the  bar  (§  269).  The  mass  of 
unit  length  is  known,  the  specific  heat  is  also  determined,  and  the 
product  of  these  quantities  into  the  rate  of  fall  of  temperature  gives 
the  rate  of  loss  of  heat.  This  being  known  for  all  the  various 
temperatures  observed  at  the  different  parts  of  the  bar  in  the  first 
experiment,  the  total  rate  of  loss  of  heat  from  the  portion  of  the 
large  bar,  beyond  any  given  section,  is  easily  calculated.  And  so 
the  conductivity,  at  particular  temperatures,  can  be  found. 

In   the  second  experiment  the  temperature  of  the  air  is  obtained 


348  A  MANUAL  OF  PHYSICS. 

by  means  of  the  long  bar,  so  that  the  results  in  both  cases  can  be 
compared,  as  is  necessary,  at  the  same  excess  of /temperature  over  that 
of.  the  surrounding  air.  But  in  addition  to  this,  since  the  rate  of 
cooling  depends  upon  temperature  and  pressure  of  the  air,  it  is 
necessary  to  perform  both  experiments  under  as  nearly  as  possible 
the  same  conditions  of  temperature  and  pressure. 

The  unit  of  heat  which  is  employed  in  this  method  is  obviously 
the  amount  of  heat  which  is  required  to  raise  the  temperature  of 
unit  volume  of  the  substance  by  one  degree,  for  the  amount  of  the 
heat  is  measured  in  terms  of  changes  of  temperature  in  the  bar.  Con- 
sequently the  quantity  which  is  so  determined  is  not  the  therma 
conductivity  as  above  denned.  Maxwell  calls  it  the  Thermometric 
Conductivity;  Thomson  calls  it  the  Thermal  Diffusivity.  The 
thermal  conductivity  of  any  substance  is  obviously  the  product  of 
the  thermometric  conductivity  into  the  thermal  capacity  of  that 
substance. 

Tait,  who  repeated  and  extended  Forbes'  experiments,  gives  the 
following  values  of 

Thermometric  Conductivity. 

Temperature  C.                          0°.                100°.  200°.  300°. 

Iron            ...         0-0149        0*0128  0*0114  0*0105 

Copper,  electrically  good . . .     0'076          0*079  0*082  0-085 

Copper,  electrically  bad   . . .     0'054           0-057  0'060  0*063 

German  silver       0*0088        0-009  0*0092  0-0094 

This  table  indicates  that,  with  the  exception  of  that  of  iron,  the 
thermometric  conductivity  of  all  these  substances  increases  as  the 
temperature  rises. 

Tait  also  gives,  for  the  iron  and  the  two  specimens  of  copper  (the 
units  being  the  foot,  the  minute,  and  the  degree  C.),  the  following 
values  of  *** 

Thermal  Conductivity. 

Iron     0-788(1-0-00002^) 

Copper,  electrically  good      ...     4'03    (1+0'0013£) 
Copper,  electrically  bad        ...     2-84    (1+0-00140 

It  appears,  therefore,  that  in  general  the  thermal  conductivity 
increases  as  the  temperature  rises.  Also,  the  order  of  the  metals 
with  regard  to  conduction  of  heat  is  the  same  as  their  order  with 
regard  to  conduction  of  electricity.  Forbes  had  observed  this  fact, 
and  had  expected  that,  as  in  the  case  of  electric  conduction,  the 
thermal  conductivity  would  decrease  as  the  temperature  becomes 
higher.  This,  as  we  see,  does  not,  in  general  at  least,  hold  true. 


CONDUCTION   AND   CONVECTION   OF   HEAT.  349 

Dr.  A.  C.  Mitchell  has  recently,  under  Professor  Tait's  direction, 
repeated  these  experiments  with  the  same  bars,  but  under  improved 
conditions.  For  one  thing,  all  the  bars  were  nickel-plated,  so  as  to 
avoid  alterations  of  the  surface  from  oxidation  at  high  temperature. 
His  results  are,  on  the  whole,"  confirmatory  of  the  previous  results 
— with  this  chief  exception,  that  he  found  the  temperature  co-efficient 
for  iron  to  be  positive,  as  it  is  in  all  the  other  substances. 

A  different  method  was  employed  by  Angstrom.  In  his  method 
one  end  of  the  bar  is  alternately  heated  and  cooled  during  equal 
periods  of  time,  the  temperature  of  the  source  of  heat  being  kept 
constant.  The  alternations  of  heating  and  cooling  are.  maintained 
until  all  the  thermometers  indicate  practically  periodical  changes  of 
temperature.  Fourier's  mathematical  investigations  show  that,  if 
the  variations  of  temperature  do  not  sensibly  affect  the  conductivity 
and  the  specific  heat,  the  conductivity  can  be  calculated  from  the 
rate  at  which  the  range  of  temperature  diminishes  per  unit  of  length 
of  the  bar,  together  with  the  observed  speed  at  which  the  '  waves  of 
temperature  '  run  along  the  bar,  provided  that  the  rate  of  surf  ace - 
loss  of  heat  is  proportional  to  the  excess  of  the  temperature  of  the 
bar  over  that  of  its  surroundings. 

285.  Conduction  through  the  Earth's  Crust.  —  Angstrom's 
method  has  a  direct  application  to  the  problem  of  the  conduction  of 
the  diurnal  and  annual  waves  of  solar  heat  downwards  through  the 
crust  of  the  earth.  In  this  investigation  we  may  assume  that  the 
heated  surface  is  practically  an  infinite  plane,  and  that  the  flow  of 
heat  takes  place  in  lines  perpendicular  to  this  plane. 

Let  ab  (Fig.  161)  represent  the  surface,  and  let  cd,  ef,  represent 
planes  parallel  to  the  surface,  at  distances  x  and  x-\-dx,  respectively, 
from  it.  If  c  be  the  thermal  capacity  of  the  substance  through  which 


ct 


f 

FIG.  161. 

the  flow  takes  place,  while  v  is  its  temperature,  and  t  represents 
time,  the  quantity  of  heat  which  enters  a  portion  of  the  substance, 
of  small  thickness  dx  and  area  a,  in  a  small  interval  of  time  dt  is 

JU. 

....  (1) 


350  A   MANUAL    OF    PHYSICS. 

for  cadx  represents  the  quantity  of  heat  which  must  be  abstracted 
in  order  to  lower  the  temperature  of  the  volume  aSx  of  the  substance 
by  one  degree,  and  dv/dt  .  £t  is  the  change  of  temperature  in  the 
time  Si. 

If  Tc  be  the  conductivity  of  the  substance,  the  quantity  of  heat 
which,  in  the  time  St,  crosses  in  the  positive  direction  the  area  a  of 
the  side  of  the  slab  nearest  the  surface  is 


dx 

Similarly,  the  quantity  which  passes  downwards  through  the  area  a 
of  the  surface  which  is  distant  from  the  former  by  the  amount  dx  is 
(since  dx  is  small) 

7  dv  t    d  (-.  dv\ 

&j--Hjn&j-  < 

dx     dx\  dx) 

Consequently,  the  amount  of  heat  which,  on  the  whole,  enters  the 
volume  a&x  in  the  time  dt  is 


and  hence  we  get 

c~v=  —(%—}        (3) 
db    dx\  dx)'  ' 

since  each  of  the  quantities  (1)  and  (2)  represents  the  same  amount 
of  heat. 

Now  (§§  27,  67)  let  us  consider  this  equation  simply  as  an  equa- 
tion of  dimensions.  The  difference  of  temperature  dv  appears 
linearly  on  both  sides  of  (3),  and  therefore  the  range  of  temperature 
does  not  appear  in  the  dimensional  equation.  We  get 

C_JL 

T~W ( ' 

where  t  represents  time,  and  I  represents  length  ;  for  dx,  the  dimen- 
sions of  which  are  those  of  a  length,  appears  twice  as  a  factor  in  the 
denominator  on  the  right-hand  side  of  (3)  ;  while  &  and  dt  appear 
once  only  on  the  right-hand  side  and  the  left-hand  side,  respectively, 
of  that  equation ;  and  we  must  remember  that  the  sign  of  equality 
indicates  now  equality  of  dimensions  alone. 
From  (4)  we  get 


M 

=  V    c' 


CONDUCTION    AND    CONVECTION    OF    HEAT.  351 

which  means  that,  if  the  times  are  altered  in  any  fixed  proportion 
p,  the  lengths  must  be  altered  in  proportion  to  the  square  root  of  p 
in  order  that  the  flow  of  heat  may  take  place  under  similar  condi- 
tions in  the  altered  circumstances.  In  other  words,  the  distances  at 
which  similar  effects  are  felt  (for  example,  the  distances  below  the 
surface  at  which  the  periodic  variations  of  surface-temperature 
cease  to  be  felt]  are  proportional  to  the  square  root  of  the  period. 

Now  the  period  of  the  annual  heating  and  cooling  is  365  times  as 
great  as  the  period  of  diurnal  variations.  Hence  the  effect  of  the 
summer's  heat  is  felt  about  nineteen  times  as  far  below  the  surface 
as  the  effect  of  the  diurnal  heat  is  felt. 

Again, 


r 

r  v  ct' 


Here  we  may  suppose  that  I  represents  the  length  of  a  wave,  while 
t  is  the  periodic  time,  so  that  the  fraction  on  the  right  hand  is  pro- 
portional to  the  rate  at  which  the  wave  of  heat  travels  downwards. 
We  see,  therefore,  that  this  rate  is  directly  proportional  to  the  square 
root  of  the  conductivity,  and  is  inversely  proportional  to  the  square 
root  of  the  thermal  capacity  and  the  periodic  time  conjointly. 

It  follows  that,  when  the  period  is  constant,  the  date  at  which 
the  maximum  temperature  reaches  any  given  depth  is  later  than 
the  date  at  which  it  left  the  surface  in  direct  proportion  to  the 
depth. 

The  law  which  regulates  the  diminution  of  the  range  of  tempera- 
ture with  increase  of  depth  cannot  be  obtained  from  equation  (3)  in 
the  way  in  which  we  have  obtained  the  two  laws  just  enunciated  ; 
for  the  temperature  does  not  appear  in  equation  (4).  But  we  may 
write  (3)  in  the  form 


^dvdx_  d  (->dv\ 
"dxdt  ~dx(  dx)  ' 


and  we  may  suppose  that  dxjdt  is  the  speed  with  which  the  heat- 
wave travels  downwards,  in  which  case  the  equation  becomes 


V 


ck     dv_  d/,dv 
T  '  dx- 


where  T  is  the  periodic  time,  and  dv/dx  now  represents  the  rate  at 
which  (say)  the  maximum  temperature  changes  as  the  wave  passes 
down. 

This  equation  asserts  that  the  rate  of  diminution  of  the  rate  of 
change  of  temperature  with  depth  is  proportional  to  the  rate  of 
change  itself.  In  other  words,  the  rate  of  change  diminishes  in 


352  A    MANUAL    OF    PHYSICS. 

geometrical  progression  as  the  depth  increases  in  arithmetical  pro- 
gression. And  its  rate  of  diminution  is  */cj  v/^T.  But,  since  the 
rate  of  diminution  of  the  rate  of  alteration  of  the  range  is  propor- 
tional to  the  rate  of  alteration  itself,  it  follows  that  the  rate  of  altera- 
tion bears  the  same  ratio  to  the  range.  Hence  the  range  diminishes 
in  geometrical  progression  as  the  depth  increases  in  arithmetical 
progression,  the  rate  of  diminution  being  directly  as  the  square 
root  of  the  thermal  capacity,  and  inversely  as  the  square  roots  of  the 
conductivity  and  the  periodic  time  conjointly. 

As  the  result  of  direct  observations  (begun  by  Forbes  in  Edinburgh 
in  1837)  of  the  temperature  at  different  distances  below  the  surface 
of  the  earth,  it  is  found  that  the  annual  heat-wave  travels  inwards 
at  the  rate  of  little  more  than  sixty  feet  per  annum,  and  that  the 
range  of  temperature  has  diminished  to  a  very  small  fraction  of  its 
original  amount  when  half  of  that  distance  has  been  traversed. 
The  diurnal  heat  is  therefore  inappreciable  at  a  depth  of  at  most 
about  two  feet.  Of  course,  all  these  results  depend  upon  the  nature 
of  the  soil. 

The  thermometers  nearest  the  surface  are  affected  by  changes  in 
the  weather,  but  these  disturbances  rapidly  die  out. 

When  a  steady  state  of  temperature  is  reached,  dv/dt  vanishes, 
and  (3)  becomes 

dfdv 
'dx\  da 

This  gives  dv 

K-=—  =  a  (constant), 
dx 

which  shows  that  the  temperature-gradient  varies  inversely  as  the 
conductivity.  This  applies  directly  to  the  case  of  the  earth  regarded 
as  a  cooling  body,  and  shows  us  that  in  strata  throughout  which  the 
conductivity  does  not  vary,  the  temperature  increases  uniformly 
per  unit  of  depth.  Of  course,  if  the  earth  is  regarded  as  a  cooling 
body,  the  steady  state  of  temperature  is  impossible ;  but  its  rate  of 
cooling  is  so  slow  that  the  time -variations  of  temperature  may 
be  neglected. 

Fourier's  equations,  when  applied  to  past  time  as  regards  the 
earth  or  the  sun,  indicate  a  state  of  uniform  high  temperature 
throughout  the  mass — a  state  which  could  not  have  arisen  by  any 
process  of  conduction.  This  suggests  the  production  of  the  heat  by 
the  gravitation  of  separate  masses  (§  93). 

286.  Conduction  in  Crystalline  Bodies.  —  Crystalline  bodies 
possess,  in  general,  unequal  conducting  power  in  different  directions. 


CONDUCTION   AND   CONVECTION   OF   HEAT.  353 

The  conducting  power  is  symmetrical  with  regard  to  three 
rectangular  axes — called  the  principal  axes  of  thermal  conduc- 
tivity. 

If  a  single  point- source  of  heat  were  placed  in  the  interior 
of  a  crystal,  the  loci  of  constant  temperature  would  be  con- 
centric ellipsoids  surrounding  that  point ;  and  Stokes  has  shown 
that  the  conductivities  parallel  to  the  axes  of  these  ellipsoids  ar£ 
proportional  to  the  squares  of  the  axes.  Sections  of  such  an 
ellipsoid  can  be  obtained  by  means  of  thin  plane  plates  of  the  crystal 
cut  in  different  directions  from  the  substance.  If  a  copper  wire  be 
passed  through  a  small  hole  drilled  through  such  a  plate  in  the 
direction  of  the  axis  of  the  ellipsoid  conjugate  to  the  plane,  and  if 
this  wire  be  heated  by  an  electric  current,  the  heat  conducted  away 
by  the  plate  may  be  made  to  melt  a  thin  coating  of  beeswax  on  the 
surface  of  the  plate.  The  boundary  between  the  melted  and  the 
unmelted  wax  is  a  section  of  the  above  ellipsoid. 

287.  Conduction  in  Liquids  and  Gases. — The  thermal  conduc- 
tivities of  liquids  (neglecting  liquid  metals)  are  small  in  comparison 
with  those  of  solids  ;  and  those  of  gases  are  smaller  still. 

In  experimental  investigations  on  the  subject,  great  care  must  be 
taken  to  avoid  convection  currents  (see  below),  which  would  com- 
pletely invalidate  the  results. 

In  the  case  of  gases,  the  conductivity  can  be  calculated  from  the 
kinetic  theory. 

288.  Convection. — Under  gravity,  all  liquids  and  gases  tend  to 
arrange  themselves  in  horizontal  layers  the  densities  of  which  de- 
crease as  their  distances  from  the  earth's  surface  increase.     This 
condition  may  be  entirely  disturbed  because  of  variations  of  tem- 
perature ;   for  the  consequent  changes  of  density  destroy  the  equili- 
brium, and  currents  are  set  up  in  the  fluid  so  as  to  restore  it.    These 
are  called  '  convection-currents.' 

We  have  already  discussed  a  typical  case  in  dealing  with  the 
maximum  density  point  of  water. 

Very  marked  examples  occur  on  a  large  scale  in  nature.  The 
trade  -winds  are  due  to  the  ascent  of  hot  air  currents  in  equatorial 
regions,  while  colder  air  blows  in  from  the  polar  regions  to  take  its 
place.  [The  north-easterly  or  south-westerly  direction  of  these  winds 
is  due  to  the  rotation  of  the  earth.]  A  considerable  part,  at  least,  of 
ocean  circulation,  is  also  of  the  nature  of  convection.  Again,  when- 
ever water  evaporates,  heat  is  absorbed  to  be  evolved  wherever 
condensation  occurs.  This  is  indeed  one  of  the  most  fruitful  sources 
of  violent  storms ;  for,  if  sufficient  heat  is  developed,  the  consequent 
increase  of  temperature  causes  a  rapid  up-rush  of  air,  and  so  creates 

23 


354  A   MANUAL    OF    PHYSICS. 

a  partial  vacuum  which  gives  rise  to  a  violent  inflow  of  the  sur- 
rounding air.  [The  air  which  comes  from  the  south  has  (in  the 
northern  hemisphere)  a  greater  eastward  motion  than  that  which 
comes  from  the  north,  and  so  a  counter-clockwise  vortex  motion  is 
produced.  Thus  the  rotation  of  a  cyclone  is  explained.] 

Practical  applications  of  the  principles  of  convection  are  seen  in 
the  usual  methods  of  boiling  water,  of  ventilation,  etc. 

A  hot  body,  which  is  cooling  in  a  gas  or  a  liquid,  loses  heat  by 
convection  as  well  as  by  radiation.  The  law  of  convective  cooling 
in  a  gas  has  been  elaborately  studied  by  Dulong  and  Petit.  Their 
results  are  expressed  by  the  formula 


where  r  is  the  rate  of  cooling,  a  is  a  constant  for  a  given  gas  and  a 
given  body,  p  is  the  pressure  of  the  gas,  6  is  a  constant  for  any  one 
gas,  and  9  is  the  excess  of  the  temperature  of  the  cooling  body  over 
that  of  the  gas.  The  rate  is  independent  of  the  nature  of  the  surface 
of  the  body  but  varies  with  its  form  and  dimensions. 


CHAPTER  XXV. 

THERMODYNAMICS  :    HEAT   AND   WORK. 

289.  Mechanical  Equivalent  of  Heat.  First  Law  of  Thermo- 
dynamics.— Heat,  since  it  is  &  form  of  energy,  may  be  transformed 
into  mechanical  work  and  into  all  other  forms  of  energy ;  but  the 
former  transformation  is  the  only  one  with  which  we  are  at  present 
concerned. 

Colding  and  Joule  were  the  first,  after  Rumford,  to  make  deter- 
minations of  the  amount  of  work  which  can  be  produced  from  a  given 
amount  of  heat,  i.e.,  of  the  mechanical  (or,  more  properly,  the  dyna- 
mical) equivalent  of  heat. 

The  most  direct  method  of  conducting  such  an  investigation  con- 
sists in  spending  a  known  amount  of  work  in  the  production  of  heat 
by  friction.  This  method  was  used  by  Joule,  who  caused  a  falling 
weight  to  drive  a  vane  rotating  in  the  interior  of  a  calorimeter 
which  contained  a  known  amount  of  water.  The  amount  of  heat 
developed  was  determined  by  means  of  the  observed  increase  of  the 
temperature  of  the  water,  due  precaution  being  taken  to  correct  for 
the  loss  of  heat  by  radiation,  and  for  the  heat  developed  by  friction 
between  the  parts  of  the  apparatus.  Various  other  methods  were 
used  by  Joule,  Him,  Regnault,  and  others.  For  example,  Joule 
proved  experimentally  that  the  heat  developed  by  the  sudden  com- 
pression of  air  is  practically  equivalent  to  the  work  spent  in  com- 
pression (§  294) ;  and  from  this  result,  together  with  an  accurate 
determination  of  the  specific  heats  of  air,  he  obtained  the  value  of 
the  mechanical  equivalent.  He  also  determined  its  value  by  means 
of  the  heat  developed  in  a  conductor  by  the  passage  of  a  current  of 
electricity  through  it  under  given  conditions  (§  342).  One  of  Hirn's 
series  of  experiments  was  made  upon  a  heat-engine  hi  actual  use. 
In  another  series,  he  experimented  upon  the  heat  developed  by  per- 
cussion. The  latter  of  these  gave  a  good  result ;  the  former  did 
not. 

Joule  finally  gave  the  number  772  (in  foot-pounds  at  the  latitude 

23—2 


356  A   MANUAL   OF   PHYSICS. 

of  Manchester)  as  the  amount  of  work  necessary  to  raise  the  tem- 
perature of  one  pound  of  water  by  one  degree  Fahrenheit.  The 
equivalent  of  the  heat-unit  (Centigrade)  which  we  have  hitherto 
used,  is  therefore  1,390  foot-pounds. 

These  experiments  prove  the  law  of  conservation  of  energy  in  so 
far  as  heat  and  work  are  concerned.  The  statement  of  the  law  of 
conservation  for  these  two  forms  of  energy  goes  by  the  name  of 
the  FIRST  LAW  OF  THERMODYNAMICS,  which  asserts  that  wJien 
equal  quantities  of  mechanical  effect  appear  from  purely  thermal 
sources,  or  disappear  in  the  production  of  thermal  effects  alone, 
equal  quantities  of  heat  disappear  or  are  produced. 

290.  Carnot's  Complete  Cycle  of  Operations — Although,  as  is 
indicated  in  the  previous  section,  we  can  determine,  by  experiment, 
the  direct  relation  between  given  amounts  of  heat  and  work,  we 
are  not  entitled  to  draw  any  conclusion  regarding  the  relation 
between  the  heat  which  disappears  and  the  work  which  appears 
in  any  given  physical  process,  unless  a  certain  condition  be  observed. 
The  necessity  for  this  condition  (which  has  already  been  referred  to 
in  §  254)  was  pointed  out  by  Sadi  Carnot. 

The  condition  is  that  the  working  substance  must  pass  through  a 
complete  cycle  of  operations,  i.e.,  a  cycle  at  the  end  of  which  the 
substance  has  returned  to  its  original  physical  state.  Heat  may 
have  been  expended  in  the  given  cycle,  and  work  may  have  been 
produced ;  but,  unless  the  final  state  is  the  same  as  the  initial  state, 
we  cannot  say  that  the  work  and  the  heat  are  mutually  equivalent. 
For  example,  carbonic  acid  gas  is  heated  by  compression,  but  the  heat 
developed  is  not  the  equivalent  of  the  work  spent  in  compression, 
for  work  is  done  by  the  molecular  forces  during  the  process. 

In  order  to  be  able  to  reason  correctly  upon  the  connection  between 
heat  and  work,  Carnot  assumed  the  existence  of  a  heat-engine  which 
can  never  be  realised  in  practice.  But  this  does  not  render  his 
results  any  the  less  valuable,  for,  in  order  that  we  may  be  able  so  to 
modify  Carnot's  results  that  they  may  apply  to  the  special  case,  we 
only  require  to  know  in  what  way,  and  to  what  extent,  any  given 
engine  differs  in  its  action  from  Carnot's. 

He  assumed  that  his  engine  was  furnished  with  a  cylinder  the 
sides  and  piston  of  which  were  absolutely  impermeable  to  heat, 
while  the  bottom  was  a  perfect  conductor  of  heat.  He  assumed 
also  the  existence  of  two  bodies,  one  hot  and  the  other  cold,  the 
temperatures  of  which  were  kept  constant.  The  former  of  these 
was  to  act  as  the  source  of  heat ;  the  latter  was  to  act  as  the  con- 
denser. The  working  substance  is  supposed  to  be  placed  in  the 
cylinder  below  the  piston,  and  may  be  any  substance  whatsoever, 


THERMODYNAMICS  :     HEAT   AND   WORK.  357 

with  any  properties  whatsoever.  But,  for  the  sake  of  definiteness, 
we  may  assume  that  it  acts  as  the  steam  in  an  ordinary  engine 
would  act. 

Let  us  suppose  that  the  cylinder,  with  its  contents  at  the  tem- 
perature of  the  cold  body,  is  placed  on  a  non-conductor  of  heat. 
The  contents  will  retain  their  temperature  (to,  say)  so  long  as  the 
cylinder  remains  on  the  non-conductor,  for  the  working  substance  is 
now  surrounded  on  all  sides  by  non-conductors.  Let  the  volume 
and  pressure  of  the  substance  be  denoted  by  v0  and  p0  respectively. 
[Eemark  here  that  the  physical  condition  of  a  known  mass  of  the 
substance  is  completely  determinate  when  any  two  of  its  tempera- 
ture, volume,  and  pressure  are  given.] 

As  the  first  operation  of  Carnot's  cycle,  the  cylinder  still  re- 
maining on  the  non-conductor,  press  down  the  piston  until  the 
temperature  of  the  substance  rises  to  that  of  the  hot  body  (£1}  say) 
and  the  volume  and  pressure  become  vl  and  pl  respectively. 

As  the  second  operation  of  the  cycle,  place  the  cylinder,  with  the 
condition  of  its  contents  unaltered,  on  the  hot  body,  and  let  the 
substance  slowly  expand  until  its  volume  becomes  (say)  vz,  and  the 
pressure  becomes  p.2  (<PI).  The  expansion  must  occur  so  slowly 
that  heat  can  flow  into  the  substance  so  as  to  constantly  prevent  its 
temperature  from  being  finitely  different  from  ^. 

As  the  third  operation  place  the  cylinder  and  its  contents — again 
without  variation  of  condition — upon  the  non-conductor,  and  let 
the  working  substance  expand  until  its  temperature  falls  to  that  of 
the  cold  body.  Let  the  pressure  and  the  volume  now  be  j»3  (<^2) 
and  v.A  respectively. 

Lastly,  place  the  cylinder  upon  the  cold  body  and  slowly  press 
down  the  piston — the  temperature  of  the  contents  remaining  at  to 
until  the  volume  again  becomes  t?o,  and  the  pressure,  therefore,  again 
takes  the  value  p0. 

This  series  of  operations  obviously  satisfies  Carnot's  condition 
that  the  final  condition  of  the  working- sub  stance  shall  be  identical 
with  its  initial  condition,  i.e.,  it  forms  a  complete  cycle. 

Now  in  the  second  operation  heat  was  taken  from  the  hot  body, 
and,  in  the  fourth  operation,  heat  was  given  to  the  cold  body.  Let 
these  quantities  be  7^  and  h0  respectively. 

Also,  in  the  first  and  fourth  operations,  work  was  done  upon  the 
contents  in  diminishing  their  volume  ;  and,  in  the  second  and  third 
operations,  the  contents,  in  their  expansion,  did  work  against  the 
external  pressure.  Let  these  quantities  be  iv0  and  WL.  respectively. 
In  the  first  and  fourth  operations  the  volume  decreased  from  v%  to 
vv  and  the  temperature  was  either  equal  to  £„  or  rising  from  t0  to 


358  A    MANUAL    OF    PHYSICS. 

^  :  in  the  second  and  third  operations  the  volume  increased  from 
vl  to  v$,  and  the  temperature  was  either  equal  to  ^  or  falling  from 
fx  to  t0.  In  the  former  pair  the  temperature  was  therefore  on  the 
whole  lower  than  it  was  in  the  latter  pair.  Consequently  the 
pressure  was  higher  when  work  was  being  performed  by  the  sub- 
stance than  when  it  was  being  expended  upon  it,  and  therefore 
Wi—  WQ  is  positive.  And,  the  cycle  being  complete,  we  can  write 


where  J  is  the  multiplier  (the  mechanical  equivalent,  sometimes  called 
Joule's  equivalent  —  hence  the  letter  J)  required  to  change  the  heat 
units  into  dynamical  units.  This  equation  is  the  analytical  expres- 
sion of  the  First  Law  of  Thermodynamics. 

291.  CarnoVs  Reversible  Cycle.  —  Besides  the  idea  of  a  complete 
cycle  of  operations  Carnot  introduced  the  equally  important  and 
fruitful  idea  of  a  Reversible  Cycle.  This  is  a  cycle  which  can  be 
performed  in  the  exact  reverse  order  :  and  Carnot's  cycle  can  be  so 
performed. 

First.  Begin  with  the  cylinder  on  the  non-conductor,  the  tempera- 
ture, volume,  and  pressure  of  its  contents  being  tlt  vlt  and  pl 
respectively.  Let  the  substance  expand  until  these  quantities 
become  t0,  v0,  andp0. 

Second.  Place  the  cylinder  on  the  cold  body  and  let  the  expansion 
proceed  until  the  pressure  and  volume  become  ps  and  v3  respectively, 
the  temperature  being  still  t0. 

Third.  Place  the  cylinder  on  the  non-conductor,  and  push  down 
the  piston  until  the  temperature  rises  to  £15  the  pressure  and  volume 
becoming  p2  and  v.2  respectively. 

Fourth.  Place  the  cylinder  on  the  hot  body  and  compress  the  con- 
tents until  the  initial  conditions  are  again  attained. 

Now  in  the  first  and  second  of  these  reverse  operations  work  was 
done  by  the  substance,  while  the  temperature  had  either  the  low 
value  t0  or  was  falling  from  ^  to  t0  :  and,  in  the  third  and  fourth 
operations,  work  was  done  upon  the  substance,  while  the  tempera- 
ture remained  at  the  high  value  £1}  or  rose  from  t0  to  tlf  On  the 
whole,  therefore,  the  temperature  —  and  consequently  the  pressure  — 
had  a  higher  value  when  work  was  done  upon  the  substance  than 
when  work  was  performed  by  it.  But  heat  was  absorbed  from  the 
cold  body  in  the  second  operation,  and  was  given  to  the  hot  body  in 
the  fourth.  In  the  reverse  cycle,  therefore,  heat  has  been  pumped 
up  from  the  condenser  to  the  source  of  high-temperature  heat,  but 


THERMODYNAMICS  :    HEAT   AND   WORK.  359 

work  has  been  expended  in  the  process.  The  complete  action  is 
also  represented  by  the  equation  of  last  section. 

Carnot  reasoned  upon  the  supposition  that  heat  was  material  and 
so  believed  that  the  quantity  of  heat  which  was  absorbed  from 
the  hot  body  was  given  to  the  cold  body  in  the  direct  process,  and 
that  the  quantity  which  was  absorbed  from  the  cold  body  was 
given  to  the  hot  body  in  the  reverse  process.  He  supposed  that  the 
heat  did  work  in  the  direct  process  merely  in  being  let  down  from  a 
source  at  high  temperature  to  a  sink  at  low  temperature,  just  as 
water  does  work  in  falling  from  a  high  to  a  low  level. 

The  interpretation  of  his  result,  on  the  principle  of  conservation 
of  energy,  is  simply  that  the  excess  of  the  heat  absorbed  over  that 
emitted  is  directly  transformed  into  its  equivalent  in  work ;  and  that, 
in  the  reverse  operation,  the  heat-equivalent  of  the  work  expended, 
together  with  the  heat  absorbed  from  the  cold  body,  is  equal  to 
the  heat  given  to  the  hot  body. 

292.  Reversibility  the  Test  of  Perfection.  Second  Law  of 
Thermodynamics. — We  shall  now  discuss  one  of  the  many  important 
results  which  can  be  deduced  from  Carnot's  principles. 

Let  an  engine  be  reversible  in  the  sense  that  all  its  physical  and 
mechanical  actions  can  be  performed  in  the  exact  reverse  order. 
Such  an  engine  is  perfect  in  the  sense  that  it  is  as  perfect  as  an.y 
engine,  working  under  the  same  conditions,  can  be,  i.e.,  the  revers- 
ible engine,  taking  in  a  quantity  7ix  of  heat  at  temperature  tlt  and 
working  with  its  condenser  at  the  temperature  £0,  will  perform  as 
much  work  as  will  any  other  engine,  working  through  the  same 
range  of  temperature,  and  also  taking  in  the  quantity  of  heat  hi  at 
the  temperature  £r 

Carnot  proved  this  statement  by  showing  that,  if  it  were  not  true, 
the  perpetual  motion  would  result.  Let  M  denote  the  reversible 
engine,  and  let  N  denote  the  (supposed)  more  perfect  engine.  Make 
N  work  directly  between  the  temperatures  tL  and  tQ,  taking  in  a 
quantity  of  heat  7^,  giving  out  a  quantity  h0,  and  performing  an 
amount  of  work  W  >  w  the  quantity  of  work  which  the  reversible 
engine  would  produce  under  the  same  conditions.  Make  M  work 
backwards  between  the  same  sources  at  the  same  high  and  low  tem- 
peratures. A  quantity  of  work  w  is  all  that  has  to  be  expended 
in  order  to  make  it  take  in  the  amount  h0  of  heat  at  the  tempera- 
ture t0  andrestore  the  amount  \  to  the  source  at  the  high  temperature. 
Hence  an  excess  of  work  W  —  w  is  gained  in  the  double  process, 
while,  on  the  whole,  no  heat  has  been  transferred  either  way.  But 
this  means  the  perpetual  production  of  work  from  nothing,  which 
is  impossible.  Hence  the  reversible  engine  is  perfect. 


360  A   MANUAL    OF   PHYSICS. 

To  adapt  this  reasoning  of  Carnot  to  the  modern  ideas  of  energy, 
we  have  only  to  argue  thus  :  Work  is  performed  in  the  double 
cycle,  while  no  heat  is  on  the  whole  taken  from  the  source,  therefore 
N  must  give  less  heat  to  the  cold  body  than  M  takes  from  it,  and  so 
the  double  engine  can  only  work  by  giving  to  the  hot  body  heat 
which  it  has  taken  from  a  colder  body,  the  temperature  of  which  it 
constantly  lowers  ;  but  this  is  in  opposition  to  all  known  facts,  and 
the  denial  of  its  possibility  may  be  safely  taken  as  axiomatic. 

This  adaptation  is  due  to  Sir  W.  Thomson,  who  re-introduced 
Carnot's  work  to  the  scientific  world  when  it  had  been  long  disre- 
garded, and  who  applied  his  principles  to  the  deduction  of  thermo- 
dynamical  results  of  the  highest  importance. 

The  statement  that  an  engine  which  is  reversible,  in  the  sense 
that  all  its  physical  and  mechanical  actions  are  capable  of  exact 
reversal,  converts,  under  given  conditions,  the  greatest  possible 
fraction  of  the  heat  which  is  supplied  to  it  into  useful  work  is  known 
as  the  SECOND  LAW  OF  THERMODYNAMICS.  We  shall  find  in  §  298 
that  its  analytical  expression  is  2(hjt)  =  o. 

293.  Absolute  Temperature.  —  The  efficiency  of  a  heat-engine  is 
the  ratio  of  the  quantity  of  heat  which  it  utilises  in  the  form  of 
mechanical  work  to  the  total  quantity  of  heat  which  is  supplied  to 
it.  In  the  notation  used  above  it  is 


The  value  of  this  fraction  is  a  maximum,  under  any  given  con- 
ditions, when  a  reversible  engine  is  used.  And  all  reversible 
engines,  whatever  may  be  the  nature  and  properties  of  the  working- 
substance,  are  equally  perfect  (i.e.,  they  possess  the  same  efficiency) 
when  they  work  through  the  same  range  of  temperature.  This 
enables  us,  as  Thomson  pointed  out,  to  obtain  an  absolute  measure 
of  temperature  —  absolute  in  the  sense  that  it  does  not  depend  upon 
the  properties  of  any  particular  substance. 

The  definition  which  Thomson  finally  adopted  was  framed  so  as 
to  make  the  absolute  scale  coincide  as  nearly  as  possible  with  the 
scale  of  the  air-thermometer.  It  is  this  : 

'  The  temperatures  of  two  bodies  are  proportional  to  the  quan- 
tities of  heat  respectively  taken  in  and  given  out  in  localities  at 
one  temperature  and  at  the  other,  respectively,  by  a  material 
system  subjected  to  a  complete  cycle  of  perfectly  reversible  thermo- 
dynamic  operations,  and  not  allowed  to  part  with  or  take  in  Jieat 
at  any  other  temperature;  or,  the  absolute  values  of  two  tempera- 
tures are  to  one  another  in  the  proportion  of  the  heat  taken  in  to 


THERMODYNAMICS  :     HEAT    AND    WORK. 


361 


the  heat  rejected  in  a  perfect  ther  mo  dynamic  engine,  working 
with  a  source  and  refrigerator  at  the  higher  and  lower  of  the 
temperatures  respectively.'1 

Expressed  in  symbols,  this  gives 


= 
A)     V 

where  ^  and  t0  are  the  absolute  temperatures  of  the  source  and  the 
condenser,  respectively. 

294.  The  Indicator  Diagram.  —  The  indicator  diagram  was 
introduced  by  Watt  for  purely  practical  purposes.  It  is,  neverthe- 
less, as  we  shall  shortly  see,  susceptible  of  numerous  important 
applications  in  pure  science. 

The  diagram  exhibits  the  relation  between  the  pressure  and  the 
volume  of  the  working  substance  used  in  any  heat  engine  ;  and 
hence  it  shows  the  work  done  in  a  complete  stroke  of  the  engine. 


aMN 

FIG.  162. 

Let  the  curve  APQBK  (Fig.  162)  represent  the  relation  between 
the  pressure  and  the  volume  at  all  stages  of  the  stroke — pressure 
and  volume  being  respectively  measured  along  the  rectangular  axes 
op  and  ov.  Let  the  co-ordinates  of  the  point  P  be  PM=p,  OM.  =  v ; 
and  let  those  of  Q  be  QN=p',  ON^v'.  When  P  and  Q  are 
indefinitely  near  each  other,  the  product  (p+p'}  (v'-v)  represents 
twice  the  area  PMNQ.  But,  neglecting  small  quantities  of  the 
second  order,  half  this  product  may  be  written  p(v'—v}.  This 
latter  product  therefore  represents  the  elementary  area  PMNQ. 

Now  the  work  done  by  a  force  /,  acting  through  a  distance  s,  is 
( §  62)  fs ;  and  the  whole  force  which  acts  upon  the  piston  in  the 
cylinder  of  the  engine  is  pa,  where  a  is  the  area  of  the  piston,  and 
p  is  the  pressure  per  unit  area,  so  that  pas  represents  the  work  done 


362  A    MANUAL    OF    PHYSICS. 

when  the  piston  moves  through  the  small  distance  s  under  the 
action  of  the  pressure  pa.  But  as  is  the  small  change  of  volume  of 
the  contents  of  the  cyclinder  which  is  produced  by  that  pressure  ; 
and  therefore  the  work  done  is  represented  by  the  area  PMNQ. 

Let  A  be  the  point  at  which  the  volume  has  its  smallest  value,  v0, 
and  let  B  be  the  point  of  maximum  volume,  vlt  As  the  volume 
expands  from  v0  to  vv  the  state  of  the  substance  being  represented  by 
a  point  which  moves  along  the  path  APQB,  the  work  performed  is 
represented  by  the  area  Aa&BQPA.  Similarly,  when  the  volume 
is  diminished  from  v1  to  -y0,  the  point  moving  from  B  to  A  along  the 
path  BRA,  the  work  expended  in  producing  compression  is  repre- 
sented by  the  area  BRAa&B.  The  difference  of  these  two  areas, 
viz.,  the  curvilinear  area  APQBRA,  therefore  represents  the  work 
which  is  expended,  on  the  whole,  during  a  complete  stroke  of  the 
engine.  Consequently,  when  the  path  is  described  in  the  positive 
direction,  work  is  expended  on  the  whole.  This  case  corresponds  to 
the  reverse  working  of  a  reversible  engine. 

When  the  closed  path  is  described  in  the  negative  direction,  work 
is  performed  on  the  whole  by  the  engine  to  an  extent  which  is  repre- 
sented by  the  total  area  enclosed  by  the  path.  [The  actual  path  might 
consist  of  a  number  of  closed  loops.  In  this  case,  each  loop  is  to 
be  considered  separately,  and  the  sum  of  the  areas — each  with  the 
proper  sign  attached,  according  as  it  is  described  positively  or  nega- 
tively—is to  be  taken  in  order  to  estimate  the  total  amount  of  work 
performed.] 

295.  Applications  of  the  Indicator  Diagram. — We  shall  now 
consider  the  application  of  the  diagram  to  the  discussion  of  the 
working  of  Carnot's  engine. 

For  this  purpose  we  must  give  the  closed  curve  a  special  form. 
In  the  second  and  fourth  direct  operations  of  Carnot's  cycle  (§  290), 
the  substance  was  maintained  at  constant  temperature  ;  in  the  first 
and  third,  no  heat,  as  such,  was  allowed  to  pass  out  of  it  or  to  enter 
it.  If,  therefore,  the  points  D,  A,  B,  C  (Fig.  163)  represent  respect- 
ively the  values  (p0,  VQ  ;  plt  Vi ;  p.2,  v.2 ;  ps,  u<)  of  the  pressure  arid 
volume  of  the  substance  at  the  commencement  of  the  first,  second, 
third,  and  fourth  operations  respectively,  all  points  on  the  line  DA 
(which  represents  the  varying  state  of  the  substance  as  it  passes 
from  the  state  (p01  t>0,  t0)  to  the  state  (plt  vlt  tj  are  characterised  by 
the  condition  that  no  heat,  as  such,  enters  or  leaves  the  working  sub- 
stance ;  all  points  on  the  line  AB  represent  states  in  which  heat  is 
absorbed  during  the  passage  from  the  condition  A  to  the  condition 
B  in  order  that  the  temperature  may  retain  the  value  ^  ;  all  points 
on  the  line  BC  represent  states  in  which,  again,  no  heat  enters  or 


THERMODYNAMICS  :     HEAT    AND    WORK. 


363 


leaves  the  substance  ;  and  all  points  on  the  line  CD  represent  states 
in  which  the  temperature  of  the  substance  is,  by  disengagement 
of  heat,  maintained  at  the  value  t0. 

Lines  such  as  AB  or  CD  are  therefore  called  "isothermals,  while 
lines  such  as  DA  or  BC  are  called  adiabatics.  ,  The  latter  name, 
which  is  due  to  Rankine,  simply  implies  that  no  heat  is  absorbed  or 
emitted. 

But  although  heat,  as  such,  neither  enters  nor  leaves  the  sub- 
stance in  the  adiabatic  condition,  mechanical  work  is  performed  or 


FIG.  163. 

expended ;  and  so,  by  transformation,  the  amount  of  heat  contained 
in  the  substance  is  actually  varying. 

If  the  amounts  of  heat  contained  in  the  substance,  in  the  states 
A  and  D,  were  identical,  and  if  the  amounts  in  the  states  B  and  C 
were  identical,  the  amounts  of  heat  which  leave  the  substance  in 
the  processes  represented  by  AB  and  DC  would  necessarily  be 
equal.  But  we  know  that  these  quantities  are  (by  our  definition)  in 
the  ratio 


tj_  and  t0  being  absolute  temperatures. 

Now  ABCD  represents  the  work  which  is  performed  in  the  direct 
cycle.  It  therefore,  by  the  principle  of  conservation  of  energy, 
represents  the  quantity  of  heat  7£1  —  7t0.  And,  if  we/arrange  matters 
so  that  7ii-7z0  =  l,  equation  (1)  shows  not  only  that  t±  — 10  m^islQbe 
equal  to  unity,  b^a^gp^that  hi  =  tlt  and  h0=t0.  V 

Let  us  therefore  intersect  the  diagram  (Fig.  164)  by  a  series  of 
isothermals  A! A2A3...,  BiB2B3...,  etc.,  and  by  a  series  of  adiabatics 
AiBiCj...,  A2B2C2...,  etc. ;  and  let  the  temperature  corresponding  to 
.  be  one  degree  below  that  corresponding  to 


364 


A   MANUAL    OF   PHYSICS. 


while  the  adiabatics  are  so  arranged  that  the  magnitude  of  each  area, 
such  as  A1A2B0B1,  is  unity.  If  we  could  continue  this  construction 
down  to  the  absolute  zero  of  temperature,  each  area  included  between 
any  isothermal,  the  isothermal  of  absolute  zero,  and  any  two  con- 
secutive adiabatics,  would  be  numerically  equal  to  the  temperature 
indicated  by  the  higher  isothermal. 


FIG.  164. 

The  experimental  data  necessary  to  the  correct  completion  of  the 
diagram  are  wanting,  but  its  correct  completion  is  not  necessary  to 
our  present  purpose.  The  following  method  of  completing  it  is  due 
to  Maxwell. 

Let  us  suppose  that  CiC2C^ ...  is  the  lowest  isothermal  whose  form 
is  correctly  known.  Draw  any  line  K1K2K3...to  represent  the 
isothermal  of  absolute  zero,  and  complete  the  adiabatics  in  such 
a  way  that  each  of  the  areas  C^B^Co,  C3K2K3Cj,,  etc.,  is  numerically 
equal  to  the  temperature  to  which  the  line  C^C.^ ...  corresponds. 
In  order  to  determine  the  position  of  the  absolute  zero,  it  is  only 
necessary  to  find  the  ratio  of  any  two  areas  such  as  AjKjE^A,,  and 
CiKxKgCa.  The  method  by  which  Thomson  and  Joule  solved  this 
problem  is  described  in  §  303. 

296.  Applications  of  the  Indicator  Diagram.  Entropy. — The  one 
set  of  lines  with  which  we  have  intersected  the  diagram  are 
characterised  by  constancy  of  temperature.  The  other  set  of  lines 
are  characterised  by  the  condition  that  no  heat  shall  be  transferred 
from  the  working  substance  to  its  surroundings,  or  from  the  sur- 


THERMODYNAMICS:   HEAT  AND  WORK.  365 

roundings  to  it.  But  we  have  seen  that  heat  does  pass  from,  or 
into,  the  substance  by  transformation.  We  have  still  to  inquire, 
therefore,  What  quantity  remains  constant  during  adiabatic  expan- 
sion or  contraction  ? 

Equation  (1)  of  last  section  enables  us  to  answer  the  inquiry.     It 
gives 


whatever  values  ^  and  t0  may  have  ;  that  is  to  say,  when  the  sub- 
stance passes  isothermally  from  any  one  definite  adiabatic  state  to 
any  other  definite  adiabatic  state,  the  quantity  of  heat  which  is 
absorbed,  or  disengaged,  bears  a  definite  ratio  to  the  temperature  at 
which  the  absorption,  or  disengagement,  takes  place. 

This  constant  quantity  h/t  may  therefore  be  regarded  as  the 
amount  by  which  some  quantity,  0,  which  is  characteristic  of  the 
substance  in  the  adiabatic  state,  changes  in  passing  from  the  one 
adiabatic  condition  to  the  other  at  constant  temperature  t.  This 
suggests  the  extension  of  the  meaning  of  the  quantity  h  to  signify 
the  total  heat  contained  in  the  substance,  so  that  we  may  define  0 
by  the  equation 

H 


where  H  represents  the  total  heat. 

The  quantity  0  was  first  called,  by  Kankine,  the  Thermodynamic 
Function.  Clausius  called  it  the  Entropy,  and  this  name  has  been 
generally  adopted. 

If  one  substance  parts  with  an  amount  of  heat,  h,  at  temperature 
t,  to  another  substance  at  temperature  t0,  the  entropy  of  the  former 
substance  decreases  by  the  amount  hjtlt  and  that  of  the  latter 
increases  by  the  amount  Jijt0.  The  total  gain  of  entropy  by  the 
system  is  therefore 


The  quantity  of  heat  which  is  absorbed  in  the  second  operation  of 
Carnot's  direct  cycle  is  t^fa  —  00),  and  the  quantity  which  is  given 
out  in  the  fourth  operation  is  tQ(<pr  -  00).  The  difference  of  those 
quantities  is  (^—  t0)  (fa  -<f>0),  which  therefore  represents  the  work 
done  in  the  complete  cycle. 

297.  Applications  of  the  Indicator  Diagram.  Total,  Available, 
and  Dissipated  Energy.— We  have  no  means  of  determining 
experimentally  the  total  amount  of  energy  in  a  given  system.  But, 


366 


A   MANUAL    OF   PHYSICS. 


in  any  practical  case,  we  only  require  to  determine  the  change  of 
energy  which  takes  place  in  the  given  operations. 

The  indicator  diagram'  enables  us  to  represent  the  total  amount  of 
energy  in  a  way  which,  though  obviously  incorrect,  leads  to  a 
correct  representation  of  the  change  of  the  total  energy  of  the  system 
which  is  produced  by  a  given  change  in  its  physical  condition. 

Let  the  points  A  and  B  (Fig.  165)  represent  respectively  the  initial 
and  the  final  conditions  of  the  system,  and  let  the  path  AB  represent 
the  series  of  changes  by  which  the  final  condition  was  arrived  at. 
[The  diagram  is  constructed  so  as  to  exhibit  the  case  in  which  there 


M    N      S 
FIG.  165. 

is  both  disengagement  of  heat  and  performance  of  mechanical  work.] 
Let  BE'  represent  the  (arbitrary)  isothermal  of  absolute  zero,  and 
let  it  be  continued  so  as  to  cut  the  axis  of  volume  in  the  point  S. 

As  the  point  which  traces  out  the  diagram  moves  from  A  to  B, 
external  work,  which  is  represented  by  the  area  ABNMA,  is  per- 
formed, and  at  the  same  time,  §  295,  an  amount  of  heat  is  dis- 
engaged which  is  represented  by  the  area  ABK'BA.  The  whole 
area  AMNBB'BA,  therefore,  represents  the  total  loss  of  energy 
which  the  working  substance  has  sustained.  We  may,  therefore, 
as  Maxwell  suggested,  regard  the  areas  AMSKA  and  BNSB'B,  re- 
spectively, as  representing  the  total  amounts  of  energy  which  are 
contained  in  the  working  substance  in  the  conditions  indicated  by  A 
and  B  respectively. 

It  is  specially  to  be  noticed  that  the  amount  of  energy  which  is 
lost  in  proceeding  from  A  to  B  is  totally  independent  of  the  path 
AB,  and  depends  only  on  the  initial  and  final  conditions  of  the 
substance. 


THERMODYNAMICS  :     HEAT   AND   WORK.  367 

Let  us  assume  now,  as  a  special  case,  that  BC  represents  the 
lowest  available  temperature.  We  may  then  cause  the  substance 
to  pass  from  the  state  A  to  the  state  B  along  the  path  ACB.  The 
work  performed  will  then  be  represented  by  the  area  ACBNMA, 
which  area  therefore  represents  the  total  amount  of  energy  which  is 
available,  under  the  given  conditions,  for  the  performance  of  me- 
chanical work.  Similarly,  BCEE'B  represents  the  heat  which  is 
necessarily  given  to  the  condenser,  i.e..  the  amount  of  energy  which 
is  necessarily  dissipated  so  far  as  the  performance  of  work  by  the 
given  system  is  concerned.  The  energy  which  is  unnecessarily 
dissipated  is  represented  by  the  area  ACB. 

298.  Thermodynamic  Motivity. — We  have  seen  that,  when  a 
quantity  of  heat,  7^,  is  given  out  by  a  body  at  temperature  tlf  the 
entropy  diminishes  by  the  amount  7*1/£1.  We  may  therefore  repre- 
sent the  total  loss  of  entropy  of  a  system  which  consists  of  a  number 
of  sources  which  are  emitting  heat  at  various  temperatures  by  the 
symbol 

2 

Similarly,  if  the  heat  emitted  by  these  bodies  is  given  to  other 
bodies  included  in  the  same  system,  we  may  denote  the  gain  of 
entropy  from  this  source  by 


The  total  loss  of  entropy  is  therefore 

\Y^i_  * 
Deleting  the  suffixes,  we  may  denote  this  simply  as 


and  our  definition  of  absolute  temperature  shows  that  this  vanishes 
wlien  a  perfect  engine  is  used.  When  any  other  engine  is  used, 
heat  is  always  lost  by  conduction  or  otherwise, "so  that  the  heat 
which  is  given  to  the  condenser  (or  is  otherwise  wasted)  is  greater 
than  h0.  Hence,  in  all  actual  cases  of  the  transformation  of  heat 
into  work  we  must  write 


A 

instead  of  sf^_: 

H 


368  A   MANUAL   OF   PHYSICS. 

where  ~k  is  greater  than  unity,  as  the  proper  expression  for  the  loss 
of  entropy.     This  is  equal  to 


which  is  necessarily  negative,  since  k  is  greater  than  unity  ;  and  so 
we  prove  Clausius'  theorem  that  the  entropy  of  the  universe  tends 
to  a  maximum. 

The  amount  of  heat  utilised  by  a  perfect  heat-engine  is  (§  293) 


Or  7*,  -^. 

fi 

If  a  number  of  such  engines  work  between  the  various  parts  of  a 
complex  system,  the  heat  which  is  not  given  to  the  condenser  is 


where  2(7*,)  now  takes  account  both  of  the  heat  which  is  taken  from 
each  body  by  some  of  the  engines,  and  of  the  heat  which  is  given  to 
it  by  others. 

In  a  perfect  engine,  the  heat  which  is  not  given  to  the  condenser 
is  represented  by  2(7*,)  alone,  and  is  entirely  converted  into  work ; 
which  again  shows  us  that,  for  such  an  engine, 

t(* 


must  vanish.  In  all  other  cases,  2(7*,)  represents  the  part  of  the 
heat  which  is  utilised  in  the  performance  of  mechanical  work,  and 
the  second  term  (which  we  must  remember  is  necessarily  positive) 
represents  the  portion  which  is  unnecessarily  wasted.  The  quantity 


therefore  represents  the  heat  which  is  dissipated  in  the  process. 

Thomson  has  tailed  the  total  energy  which  could  be  made  avail- 
able for  mechanical  work  by  a  perfect  engine  under  given  condi- 
tions, the  Thermodynamic  Motivity  of  the  system.  If  we  have  a 
medium  external  to  the  given  system,  which  may  be  used  as  a  con- 
denser, the  motivity  is  the  whole  amount  of  work  which  can  be 
obtained  by  the  perfect  engine  in  reducing  the  temperature  of  the 
system  to  that  of  the  external  medium.  If  the  engine  works  so  as 
to  equalise  the  temperatures  of  the  various  parts  of  the  system,  the 
motivity  is  the  whole  amount  of  work  which  can  be  so  obtained. 


THERMOrYNAMICS  :     HEAT    AND    WORK.  369 

Let  t0  be  the  final  temperature,  and  let  h  be  a  quantity  of  heat 
taken  from  a  body  at  temperature  t.  The  motivity,  so  far  as  this 
quantity  is  concerned,  is 


and,  to  obtain  the  total  motivity,  we  must  sum  all  such  quan- 
tities. 

The  total  energy,  e,  in  any  system  is  equal  to  the  sum  of  the 
motivity,  ra,  and  the  dissipated  energy  of  that  system.  If  0  is  the 
entropy,  9  the  temperature,  and  J  the  mechanical  equivalent  of 
heat,  the  dissipated  energy  is  J00.  Hence,  if  the  system  passes 
from  a  state  indicated  by  the  suffix  1  to  a  state  indicated  by  the 
suffix  2,  we  get 

ml  -  m.2  =  el  -  ez 


No  change  can  take  place  of  itself  in  the  system  unless  thereby 
the  motivity  be  decreased  —  that  is,  unless  m1  -  m.2  be  positive.  But 
m^—m^  may  be  positive,  although  el  —  e%  is  negative,  provided  that 
0.2<£2  is  sufficiently  greater  than  0^.  Hence  we  see  that  a  given 
chemical  action  may  take  place  of  itself  with  absorption  of  heat, 
provided  that  a  sufficient  amount  of  energy  be  dissipated  in  the 
process.  (See  §  280.) 


24 


CHAPTEK  XXVI. 

THERMODYNAMICAL    RELATIONS. 

299.  IN  the  course  of  the  discussion,  in  last  chapter,  of  the  con- 
nection between  heat  and  work,  we  were  led  to  consider  five  quan- 
tities in  terms  of  which  the  physical  condition  of  a  substance  may 
be  represented.  These  quantities  were  the  energy,  e  ;  the  entropy, 
0  ;  the  pressure,  p  ;  the  volume,  v  ;  and  the  temperature,  t. 

But  we  were  also  led  to  see  that  the  physical  condition  of  the  sub- 
stance was  completely  determined  when  two  of  these  quantities,  p 
and  v,  were  given  :  for,  on  the  indicator  diagram,  we  could  lay 
down  lines  of  constant  temperature,  of  constant  entropy,  and  of 
constant  energy.  The  total  values  of  the  two  latter  quantities  were 
not,  it  is  true,  indicated  ;  but  that  was  due  to  a  defect  in  our  know- 
ledge, and  not  to  any  defect  necessarily  inherent  in  the  diagram. 

It  at  once  follows  that  the  variation  of  any  one  of  the  five  quan- 
tities can  be  represented  in  terms  of  the  simultaneous  variation  of 
any  two  of  the  rest.  Thus  we  may  write 


.....  (1). 

de=fdt  +  gdv,  .....  (2). 

de  =  mdt+ndp,  .....  (3). 
and  so  on. 

300.  We  shall  first  consider  equation  (1).  If  the  volume  be  con- 
stant we  obtain  de  =  ad<{>.  But  we  know  that,  under  constant  volume, 
the  energy  increases  by  the  amount  (in  dynamical  units)  of  heat 
which  has  been  supplied  ;  and  we  also  know,  by  the  results  of  last 
chapter,  that  this  amount  is  td<j>.  Hence  a  =  t.  Similarly,  if  no 
heat  be  supplied,  so  that  0  remains  constant,  (1)  becomes  de  =  bdv. 
But  under  these  conditions  the  energy  diminishes  by  the  amount  of 
external  work  which  is  performed,  that  is,  by  the  amount  pdv. 
Hence  b=  -p,  and  (1)  becomes 

de  =  td<f>-pdv  .....  (4). 

/  de\         ,  (de\ 

This  gives  (—      =  t,  —)    =  -p. 

\dJ,        \&>f+ 


THERMODYNAMICAL   RELATIONS.  371 

The  suffixes  denote  respectively  that  the  volume  and  the  entropy  are 
constant  ;  so  that  the  former  equation  asserts  that,  at  constant 
volume,  the  increment  of  the  energy  is  equal  to  the  heat  supplied  ; 
while  the  latter  asserts  that,  under  adiabatic  expansion,  the  decre- 
ment of  the  energy  is  equal  to  the  amount  of  work  which  is  per- 
formed. These  equations  therefore  express  the  conditions  upon 
which  we  deduced  (4)  from  (1). 
From  them  we  get 


e_=  (dt  \ 
d(j>     \dv) 


dvd(j>     \dv  $   d<l>dv~ 
WMehgive  (£)--(*),  .......  .(5) 

where  we  may  dispense  with  the  suffixes.  If  the  right-hand  side  of 
this  equation  be  simultaneously  multiplied  and  divided  by  t,  the 
denominator  represents  the  amount  of  heat  which  is  -supplied,  at 
constant  volume,  in  order  to  produce  the  variation,  dp,  of  pressure. 
If  dp  and  td<j>  are  positive,  dtjdv  is  essentially  negative.  Hence  the 
equation  asserts  that  substances  which,  at  constant  volume,  have 
their  pressure  raised  (or  diminished)  by  the  application  of  heat, 
will  fall  (or  rise]  in  temperature  during  adiabatic  expansion  ;  and 
the  change  of  temperature,  per  unit  change  of  volume,  is  numeri- 
cally equal  to  the  product  of  the  absolute  temperature  into  the 
change  of  pressure  per  unit  of  heat  supplied. 
We  may  now  combine  with  equation  (1)  the  equation 


d(pv)  =p 
as  the  result  of  which  we  get 

d(e+pv)  =  td<l>+vdp  .........  (6). 

From  this  we  deduce,  as  above,  the  result 


If  dt  and  dp  are  both  positive,  dv  and  d<f>  are  necessarily  of  the 
same  sign.  Hence,  multiplying  and  dividing  the  right-hand  side  by 
t,  we  see  that  substances  which  expand  (or  contract),  under  constant 
pressure,  when  heat  is  supplied  to  them,  rise  (or  fall)  in  tempera- 
ture when  they  are  subjected  to  adiabatic  compression;  and  the 
change  of  temperature,  per  unit  increase  of  pressure,  is  equal  to 

24—2 


372  A   MANUAL   OF   PHYSICS. 

the  product  of  the  increase  of  volume  at  constant  pressure,  per 
unit  of  heat  supplied,  into  the  absolute  temperature. 
If  we  now  combine  with  (1)  the  equation 


we  obtain 

d(e-t<j>)=-<pdt-pdv  ........  (8). 

Therefore 


dvJ~\dtJ  " (9)> 

Multiplying  each  side  of  (9)  by  t  we  see  that  substances,  which 
absorb  (or  emit)  heat  when  their  volume  increases  isothermally, 
have  their  pressure,  at  constant  volume,  raised  (or  diminished)  by 
increase  of  temperature  ;  and  the  change  of  pressure,  per  unit  rise 
of  temperature,  is  equal  to  the  quotient  by  the  absolute  tempera- 
ture of  the  heat  which  is  absorbed  (or  emitted). 

Let  L  represent  latent  heat,  and  let  v'  —  v  represent  the  change  of 
volume  of  unit  mass  of  the  substance  when  it  changes  its  state.  In 
this  case  (9)  becomes 

t(v'-^v): 

from  which  we  can  calculate  the  change  of  the  melting-point,  or  the 
boiling-point,  which  results  from  a  given  change  of  pressure. 
Combining  (8)  with 

d(pv)  =pdv+vdp, 
we  find 

d(e-t$+pv)=  -tydt+vdp, (10), 

which  leads  to 


This  tells  us  that  substances  which  expand  (or  contract)  under  con- 
stant pressure  when  their  temperature  is  raised  emit  (or  absorb) 
heat  in  order  that  their  temperature  may  remain  constant  ivlien 
the  pressure  is  increased;  and  the  heat  which  is  evolved  (or 
absorbed)  per  unit  increase  of  pressure  is  equal  to  the  continued 
product  of  the  temperature,  the  volume,  and  the  expansibility. 
For  the  expansibility  is  Ifv  .  dv/dt. 

301.  In  equation  (2)  the  quantity/  represents  the  rate  at  which 
the  energy  increases,  per  unit  increase  of  temperature,  at  constant 
volume.  It  therefore  represents  the  specific  heat  at  constant 


THERMODYNAMICAL    RELATIONS.  373 

volume,  which  we  have  already  denoted  by  the  symbol  c.     Hence 
we  have  —  from  (1)  — 

cdt  =  td<}>, 

with  the  condition  dv  =  0. 

These  equations  may  be  written  in  the  form 


°= 

But  the  quantity 


is  evidently  the  specific  heat  at  constant  pressure,  hitherto  denoted 
by  k.     Hence  (12)  becomes 


dp  ...........  (14). 

Now  the  condition  dv=Q  necessitates  a  certain  relation  between 
•dp  and  dt  ;  but  we  can  eliminate  these  quantities  from  (13)  and  (14). 


Thus  fc-^/T^f^)- 

(dv\    \dtJp 

By  (11)  this  becomes 


To  apply  this  result  to  the  case  of  a  perfect  gas  we  must  find  the 
values  of 

(I)  -d(?) 

^dth         \dpJi 
from  the  equation  pv=Ht. 

mu.  (dv\        R         _  fdv\          Rt 

This  gives  (  -  )    =  —  ,  and  I—  )  =  -  -  , 

\dth       p  \dph         p2 

whence  fe-c  =  R. (16). 


374  A   MANUAL   OF   PHYSICS. 

The  difference  between  the  two  specific  heats  is  therefore  constant ; 
and,  since  the  values  of  both  k  and  R  can  readily  be  found  by 
experiment,  c  (the  experimental  determination  of  which  is  very 
difficult)  can  be  calculated  by  means  of  this  relation.  (See  §  271.) 

302.  The  equation  connecting  the  pressure,  volume,  and  tem- 
perature of  a  perfect  gas  is  pv  =  R£.  It  will  be  useful  to  determine 
the  relation  between  the  pressure,  the  volume,  and  the  entropy  of 
such  a  gas.  We  have 


where  td^jdt  is  obviously  the  specific  heat  at  constant  pressure, 
and,  by  (11),  —  d$ldp  is  equal  to  dv/dt  at  constant  pressure,  which 
again  is  equal  to  R/j?,  i.e.,  to  (7t  —  c)/p.  Hence 

dt     n       .dp 
^  =  &T-(&-c)  —  , 
t  p 

the  integral  of  which  (§  38)  is  /'•*  ) 

t+(1c-c)logp, 


where  a  (and  therefore  log  a)  is  a  constant.     We  may  write  this  in 
the  form 


But  t  is  equal  to  pvfR,  whence 


A  being  equal  to  aR*.     Thus,  instead  of  pv  =  constant,  we  must  write 

* 
pvc=  constant 

when  adiabatic  compression  or  expansion  takes  place. 

303.  When  air  is  compressed  by  the  sudden  application  of 
pressure,  the  heat  developed  is  almost  precisely  equivalent  to 
the  work  which  is  spent  in  producing  the  compression.  Joule 
proved  this  by  enclosing  air  in  a  strong  vessel  which  could  be 
placed  in  communication  with  another  vessel,  of  the  same  size, 
which  had  been  exhausted  of  air.  Both  vessels  were  placed  in  a 
large  mass  of  water  the  temperature  of  which  was  accurately 
determined.  When  a  stopcock  in  a  tube  connecting  the  two  vessels 
was  opened,  the  air  rushed  from  the  one  vessel  into  the  other  so  as 
to  equalise  the  pressure  throughout.  The  temperature  of  the  vessel 
containing  the  expanding  air  was  lowered,  for  work  had  been  done 
during  expansion,  so  that  the  air  was  cooled.  That  of  the  other 


THERMODYNAMICAL   RELATIONS.  375 

vessel  rose,  for  the  violent  impact  of  the  air  which  rushed  into  it 
caused  the  development  of  heat.  But  the  amount  of  heat  which 
was  absorbed  in  the  one  case  was  almost  precisely  equal  to  that 
which  was  evolved  in  the  other,  for  the  surrounding  water,  which 
was  well  stirred,  showed  no  appreciable  change  of  temperature. 

A  more  accurate  form  of  this  experiment  was  subsequently 
adopted  by  Joule  and  Thomson  in  their  researches  on  the  thermo- 
dynamical  properties  of  gases.  The  gas  under  investigation  was 
made  to  pass  very  slowly  through  a  tube,  in  which  a  plug  of  cotton 
wool  was  placed,  and  its  pressure  and  temperature  on  both  sides  of 
the  plug  were  observed. 

The  preceding  methods  lead  to  a  simple  equation  connecting 
the  changes  of  temperature  and  pressure  with  the  volume,  the  abso- 
lute temperature,  and  the  expansibility  of  the  substance.  The 
expansibility  may  then  be  expressed,  by  means  of  Charles'  Law,  in 
terms  of  the  temperature  on  the  Centigrade  scale,  if  the  range  of 
temperature  be  so  small  that  the  Centigrade  and  the  absolute 
degrees  are  practically  equal  throughout  its  extent;  so  that  the 
equation  gives  a  direct  comparison  of  the  absolute  and  the  Centi- 
grade scales. 

Boyles'  and  Charles'  Laws  give  (§  266)  for  a  perfect  gas  T  =  £ 
+  l/a,  where  T  is  absolute  temperature.  The  investigation  just 
alluded  to  gives 


where  ;//,  as  Thomson  and  Joule's  experiments  indicate,  is,  in  true 
gases,  a  small  quantity  —  generally  positive. 

The  experiments  showed  that  all  the  true  gases  except  hydrogen 
were  made  colder  by  their  passage  through  the  plug,  and  indicated 
that  the  absolute  zero  is  about  273°'7  C. 

304.  The  truth  of  the  Second  Law  of  Thermodynamics  (§  292) 
rests  entirely  on  the  immensity  of  the  number  of  particles  con- 
tained in  any  portion  of  matter  which  is  of  a  size  comparable  with 
the  dimensions  of  our  instruments  and  machines.  An  ordinary 
thermometer,  placed  in  any  position  in  a  mass  of  air,  might  indicate 
uniformity  of  temperature  ;  while  another  thermometer,  sufficiently 
small  in  size,  might  (rather,  would)  indicate  rapid  variations  of 
temperature,  and  might  even  show  that  heat  was  passing  from  cold 
parts  to  hot  parts  of  the  given  mass.  For,  the  quickly  moving  mole- 
cules might  occupy  on  the  whole  one  portion  of  a  volume  so  small 
as  to  contain  only  a  few  molecules,  while  the  slowly  moving 
molecules  occupied  the  remainder  ;  and  some  of  the  slowest  of  the 


376  A    MANUAL    OF    PHYSICS. 

quickly  moving  molecules  might  be  exchanged  for  such  of  the  slowly 
moving  molecules  as  were  actually  moving  more  quickly  than  they 
were. 

An  average  uniformity  is  preserved  on  the  large  scale,  though,  on 
a  sufficiently  small  scale,  it  does  not  obtain.  It  is  because  of  this 
average  uniformity  that  the  statement — that  a  heat-engine  cannot 
continually  draw  the  heat  which  it  transforms  into  work  from  a 
body  colder  than  its  condenser — which  Thomson  made  the  basis  of 
the  Second  Law  of  Thermodynamics  is  true. 


CHAPTER    XXVII. 

ELECTROSTATICS. 

305.  Electrification  by  Friction. — When  a  rod  of  glass  is  rubbed 
with  flannel — or,  better,  with  leather  coated  with  a  paste  of  zinc 
amalgam — it  acquires  the  property  of  attracting  surrounding  bodies. 
Light  bodies,  such  as  pieces  of  paper,  can  even  be  raised  up  by  it 
against  the  attraction  of  the  earth.  When  in  this  state,  the  glass  is 
said  to  be  electrified — or  it  is  said  that  electricity  has  been 
developed  upon  the  glass. 

If  the  glass  had  been  rubbed  with  cat's-skin  or  with  any  one  of 
several  other  substances,  similar  effects  would  have  ensued ;  but  the 
extent  to  which  electrification  is  developed  depends  upon  the  nature 
of  the  substance  which  is  used  as  a  rubber. 

The  glass  may  be  replaced  by  sealing-wax,  resin,  ebonite,  etc., 
and  the  phenomena  will  still  be  exhibited  to  a  greater  or  less 
extent. 

In  all  cases  it  is  necessary  for  success  that  the  substances  shall 
be  well  warmed  and  dried. 

306.  Conductors  and  Non- Conductors.  —  If,  instead  of  a  glass 
rod  or  a  rod  of  sealing-wax,  we  take  a  metallic  rod,  no  electrical 
effects  are  in  general  observable  ;  and  many  other  substances  also 
are  incapable  (unless  special  means  are  adopted,  §  324)  of  being 
electrified  by  friction. 

We  are  thus  led  to  divide  all  substances  into  two  classes  according 
as  they  are  or  are  not  electrifiable  by  friction  in  the  usual  way. 
Those  of  the  former  class  are  called  Insulators,  Dielectrics,  or 
Non-conductors;  those  of  the  latter  class  are  called  Conductors. 
The  latter  terms  are  applied  because  it  is  found  that  all  substances 
which  cannot  usually  be  electrified  by  friction  have  the  power  of 
allowing  electricity  to  flow  along  them,  while  the  other  class  of  sub- 
stances prevent  such  flow. 

307.  Fundamental  Phenomena  presented  by  Electrified  Bodies. 
— The  substances  which  are  attracted  by  an  electrified  body  are  not 


378  A   MANUAL   OF   PHYSICS. 

necessarily  non-conductors.  In  order  to  investigate  the  subject 
further  we  shall  suppose  that  a  pith-ball  (which  is  a  conductor, 
and  is  at  the  same  time  very  light,  so  that  the  effects  to  be  observed 
are  easily  seen)  is  the  body  to  be  attracted,  and  we  shall  suppose 
it  to  be  insulated  by  being  suspended  from  a  dry  glass  rod  by  means 
of  a  dry  silk  thread.  » 

If  an  electrified  glass  rod  be  brought  into  the  neighbourhood  of  the 
pith-ball,  the  ball  will  be  drawn  towards  it ;  and  this  will  also  take 
place  when  electrified  sealing-wax  is  presented. 

Now  let  the  glass  rod  be  brought  so  near  that  the  ball  comes  in 
contact  with  it.  Immediately  after  contact  the  ball  is  violently 
repelled  by  the  glass ;  but,  if  the  ball  be  touched  with  the  hand, 
attraction  will  again  occur ;  and  the  same  phenomena  will  happen 
when  electrified  sealing-wax,  or  any  other  electrified  body,  is  used. 
Still,  though  all  electrified  bodies  produce  this  effect,  a  slight  modifi- 
cation of  the  experiment  will  bring  to  view  a  profound  difference  in 
the  nature  of  the  electrification  of  different  substances. 

Instead  of  touching  the  pith-ball  when  it  is  repelled  by  the  glass, 
let  the  electrified  sealing-wax  be  presented  to  it.  Strong  attraction 
becomes  apparent.  Similarly  the  electrified  glass  will  attract  the 
ball  when  the  sealing-wax  repels  it.  And  all  substances  which  can 
be  electrified  by  friction  can  be  classified  according  as  they  act  in 
this  respect  like  glass  or  like  sealing-wax. 

308.  Positive  and  Negative  Electricity. — In  order  to  explain 
these  phenomena  we  make  the  following  assumptions :  1st.  There 
,are  two  '  kinds '  of  electricity ;  2nd.  Like  kinds  repel  each  other, 
unlike  kinds  attract  each  other  ;  3rd.  The  attraction  and  repulsion 
diminish  as  the  distance  increases  ;  4th.  An  unelectrified  body  may 
be' looked  upon  as  a  body  which  contains  equal  amounts  of  both 
kinds  of  electricity,  which  can  be  separated,  to  a  greater  or  less 
extent,  by  means  of  the  action  of  electrified  bodies. 

Let  us  distinguish  the  electricities  developed  on  glass  and  sealing- 
wax  as  positive  and  negative  respectively.  When  the  positively 
electrified  glass  rod  is  brought  near  to  the  unelectrified  pith-ball, 
we  assert,  in  terms  of  our  hypothesis,  that  the  neutral  electricities 
in  the  ball  are  separated,  negative  electricity  coming  to  the  side 
near  the  glass,  positive  electricity  being  repelled  to  the  far  side.  The 
attraction  between  the  unlike  kinds  is  stronger  than  the  repulsion 
between  the  like  kinds,  for  the  former  are  at  a  less  distance  apart. 
Hence  the  pith-ball  moves  towards  the  rod,  for  the  electricity  is 
confined  to  it  and  so  cannot  further  alter  its  distance  from  the  elec- 
tricity of  the  rod  unless  the  ball  moves.  If  contact  takes  place,  the 
negative  electricity,  which  has  been  induced  (as  the  phrase  is)  in 


ELECTROSTATICS.  379 

the  ball,  unites  with  a  portion  of  the  electricity  of  the  rod,  so  that 
the  ball  is  now  charged  with  positive  electricity,  and  is  therefore 
repelled  from  the  rod  until  it  loses  its  charge  (say,  by  repulsion  to 
the  ground  when  the  observer  touches  the  ball). 

The  same  reasoning,  with  the  interchange  of  the  words  positive 
and  negative,  applies  when  sealing-wax  is  used  instead  of  glass. 

Finally,  the  ball  which  has  touched  the  glass  rod  is  positively 
electrified,  and  is,  therefore,  attracted  by  the  sealing-wax  ;  and  the 
ball  which  has  touched  the  sealing-wax  is  negatively  electrified,  and 
so  is  attracted  by  the  glass  rod.  All  the  phenomena  are  thus  ex- 
plained by  means  of  our  assumptions. 

In  the  fourth  assumption  it  was  stated  that  an  unelectrified  body 
contains  equal  quantities  of  both  kinds  of  electricity.  In  accordance 
with  this  assumption,  it  may  be  proved,  by  the  methods  to  be 
shortly  described,  that  the  rubber  which  is  used  to  produce  electricity 
by  friction  becomes  electrified  to  exactly  the  same  extent  as  the 
rod  which  is  rubbed,  but  with  the  opposite  kind  of  electricity  to  that 
which  is  developed  on  the  rod. 

At  one  time  it  was  customary  to  speak  of  electricity  similar  to  that 
usually  developed  on  glass  as  '  vitreous,'  and  of  electricity  similar  to 
that  which  is  produced  on  sealing-wax  and  other  resins  as  '  resinous,' 
electricity.  The  mere  fact  that  the  so-called  resinous  electricity 
may  be  obtained  from  glass  is  sufficient  proof  of  the  undesirability 
of  this  classification.  The  terms  positive  and  negative,  as  we  have 
employed  them  above,  are  much  preferable,  for  the  words  imply 
nothing  but  a  distinction  in  kind. 

The  phrase  '  kind  of  electricity  '  is  very  apt  to  be  misleading. 
We  do  not  yet  know  what  electricity  is.  One  would  never  dream 
of  saying  that  the  resultant  positive  and  negative  forces  which  con- 
stitute a  stress  are  essentially  different  from  each  other,  or  that 
left-handed  (positive)  rotation  is  intrinsically  different  from  right- 
handed  (negative)  rotation.  Yet  equal  and  opposite  forces,  and 
equal  and  opposite  rotations,  annul  each  other's  effects.  The  terms 
'  positive  electricity  '  and  '  negative  electricity '  are  merely  adopted 
in  order  to  enable  us  to  consistently  and  concisely  describe  and  (so 
far)  explain  certain  phenomena. 

The  use  of  the  old  expression  '  electric  fluid '  is  to  be  carefully 
avoided. 

809.  The  Gold-leaf  Electroscope. — An  electroscope  is  an  instru- 
ment which  is  used  to  indicate  the  existence  of  electrification.  If, 
in  addition,  the  instrument  measures  the  magnitude  of  the  electrifi- 
cation, it  is  called  an  electrometer. 

The  gold-leaf  electroscope  is  one  of  the  most  delicate  of  all  electro- 


380 


A   MANUAL   OF   PHYSICS. 


scopes.  It  consists  of  two  pieces  of  gold-leaf  a,  a  (Fig.  166),  which 
are  connected,  by  means  of  a  metal  rod,  to  a  metal  head  h.  The  rod 
passes  through  the  top  of  a  glass  vessel  in  the  manner  indicated  in 
the  diagram.  The  glass  vessel  is  open  at  the  bottom,  and  contains  a 


FIG.  166. 

wire  cage  which  surrounds  the  gold  leaves.  The  cage  can  be 
placed  in  connection  with  the  ground  by  means  of  the  metallic  con- 
nection 6.  The  use  of  this  cage  will  appear  afterwards  (§  316) ;  in 
the  meantime  we  are  merely  concerned  with  the  manner  of  using 
the  instrument  and  the  nature  of  its  indications. 

If  a  positively  electrified  body  be  brought  into  the  neighbourhood 
of  the  head  h,  negative  electricity  is  drawn  towards  the  head,  and 
positive  electricity  is  repelled  into  the  leaves,  which  diverge,  since 
they  are  similarly  electrified.  The  closer  the  body  is  brought  to  the 
head,  the  more  widely  do  the  leaves  diverge ;  and,  when  the  body  is 
withdrawn,  they  collapse. 

If,  while  the  leaves  are  still  diverging  because  of  the  presence  of 
the  electrified  body,  the  head  h  be  momentarily  touched  by  the 
hand,  instant  collapse  of  the  leaves  will  ensue  (for  the  positive 
electricity  escapes  from  the  leaves  through  the  hand  to  the  ground), 
and  the  state  of  collapse  will  continue  so  long  as  the  electrified  body 
is  not  withdrawn.  But  when  the  body  is  withdrawn  from  the  neigh- 
bourhood of  the  head,  the  leaves  once  more  diverge ;  for  the  nega- 
tive electricity  which  was  drawn  to  the  head  of  the  instrument  now 
spreads  in  part  through  the  metal  rod  into  the  leaves.  The  latter 
therefore  are  diverging  with  negative  electricity. 

In  this  condition  the  instrument  can  be  used  to  indicate  the 
nature  of  the  electrification  of  any  body  which  is  brought  into  the 
neighbourhood  of  the  head  h.  If  the  body  be  negatively  electrified, 
more  negative  electricity  will  be  repelled  into  the  leaves  which  will 
therefore  diverge  more.  If  it  be  positively  electrified  (or  unelec- 
trified),  the  negative  electricity  is  drawn  from  the  leaves  which  then 


ELECTROSTATICS.  381 

collapse;  and  if  the  positive  electrification  be  sufficiently  strong, 
some  of  the  neutral  electricity  in  the  rod  will  be  separated,  and  the 
leaves  will  diverge  because  of  being  positively  electrified. 

The  interchange  of  the  words  positive  and  negative  in  the  above 
reasoning  will  enable  it  to  apply  to  the  case  in  which  a  negatively 
electrified  body  is  originally  brought  near  to  the  head  of  the  instru- 
ment. 

It  is  obvious  that  these  experiments  are,  in  large  part,  merely  a 
modified  repetition  of  those  which  were  discussed  in  §  307. 

310.  Electrification  by  Contact  and  by  Induction.  Electric 
Quantity. — In  §  308  we  have  spoken  of  the  electricity  which  is  in- 
duced upon  a  conducting  body  because  of  the  presence  of  another 
electrified  body.  So  long  as  contact  does  not  take  place  between 
the  two  bodies,  the  total  amount  of  induced  electrification  is  zero,  a 
certain  amount  being  drawn  to  one  side  of  the  body,  while  an  equal 
amount  of  the  opposite  kind  is  repelled  to  the  other  side.  But 
whenever  contact  occurs,  the  attracted  electricity  unites  with  some 
of  the  electricity  in  the  inducing  body ;  and  so  the  conductor  is 
electrified  with  the  same  kind  of  electricity  as  that  which  the  induc- 
ing body  possesses.  It  is  then  said  to  be  electrified  by  contact. 
The  total  effect  is  the  same  as  if  the  inducing  body  had  given  some 
of  its  electricity  to  the  conductor,  and  it  is  usual  to  say  that  it  has 
done  so ;  for  we  cannot  distinguish  one  amount  of  electricity  from 
any  other  equal  amount. 

In  the  process  of  electrification  by  contact,  the  one  body  loses  a 
certain  amount  of  electricity,  while  the  other  gains  an  equal  amount. 
This  can  be  proved  by  means  of  measurements  of  the  forces  of 
attraction  or  repulsion  which  they  exert  upon  an  electrified  body 
the  electrification  of  which  does  not  alter.  In  fact,  we  can  electric- 
ally iveigli  out  equal  amounts  of  electricity,  just  as  we  can  gravita- 
tionally  weigh  out  equal  amounts  of  matter.  We  are  therefore 
justified  in  speaking  of  electricity  as  a  thing  which  can  be  doled  out 
in  measurable  quantities ;  and  it  is  usual  to  say  that  an  electrified 
body  is  charged  with  electricity,  and  to  call  the  total  quantity  of 
electricity  which  it  possesses  its  charge. 

Suppose,  now,  that  we  have  a  charged  body — charged  positively, 
let  us  say.  It  is  possible  by  its  means  to  charge  other  bodies, 
either  positively  or  negatively,  to  any  desired  extent— and  that 
without  any  reduction  of  its  own  charge. 

Let  A  (Fig.  167)  be  the  positively  charged  body,  and  let  B  and  C 
represent  two  of  the  other  bodies,  of  wfcich  B  is  to  be  negatively 
charged,  while  C  is  to  be  positively  charged.  Each  of  the  three 
bodies  being  well  insulated  from  other  conductors,. place  B  and  C  in 


382  A   MANUAL   OF   PHYSICS. 

contact  in  some  such  position  relatively  to  A  as  is  indicated  in  the 
figure.  Then  let  B  and  C  be  separated :  B  will  be  charged 
negatively,  while  C  will  have  a  positive  charge.  Greater  effects 
would  be  produced,  if  necessary,  by  placing  B  and  C  at  a  con- 
siderable distance  apart,  and  joining  them  by  a  thin  conducting 
wire ;  for  the  effect  of  the  charge  in  A  is  largely  counteracted  by 


FIG.  167. 

the  mutual  attraction  between  the  positive  and  negative  elec- 
tricities in  B  and  C ;  and  this  mutual  attraction  is  diminished  as 
the  distance  between  B  and  C  increases.  In  practice,  it  is  con- 
venient to  let  C  be  the  earth,  and  to  place  B  in  connection  with  it 
by  means  of  a  metallic  wire  or  other  conductor.  In  this  case  the 
repelled  positive  electricity  in  C  is  practically  at  an  infinite  distance. 
We  may  then  use  the  body  B,  instead  of  A,  if  we  wish  to  charge 
any  other  body  negatively. 

This  process,  in  which  the  inducing  body  does  not  lose  any  of  its 
charge,  is  called  charging  by  induction. 

The  induced  charge  is,  except  in  one  special  case  (§  316),  less 
than  the  inducing  charge. 

311.  Continued  Production  of  Electricity.  The  Electropliorus. 
— In  last  section  we  saw  how  it  is  possible  to  obtain  a  positive  or  a 
negative  charge  at  will  by  means  of  a  single  insulated  charged 


FIG.  168. 

body  and  two  conductors.  The  instrument,  based  on  this  principle, 
which  is  generally  used  for  the  purpose,  is  the  electropliorus.  It 
consists  of  a  flat,  circular,  cake  of  resin,  contained  in  a  shallow 
metal  vessel  db  (Fig.  169).  The  resin  is  slightly  warmed,  and  is 
electrified  negatively  by  friction  with  cat's  skin.  A  metal  disc  cd, 
insulated  by  means  of  a  glass  handle,  is  used  instead  of  the  con- 


ELECTROSTATICS.  388 

ductor  B  of  last  section,  while  the  earth  takes  the  place  of  the 
conductor  C. 

As  the  metal  disc  is  brought  near  to  the  electrified  resin,  positive 
electricity  is  induced  on  its  near  side  and  negative  electricity  is 
repelled  to  its  far  side  ;  and  the  more  closely  the  disc  is  approached 
to  the  resin,  the  greater  is  the  resultant  separation  of  electricity. 

When  the  disc  is  laid  upon  the  resin,  contact  is  made  between 
it  and  the  ground  by  means  of  a  metal  pin  which  passes  through 
the  centre  of  the  cake  of  resin  and  is  connected  with  the  metal 
vessel  enclosing  it,  and  therefore  with  the  ground  upon  which  the 
vessel  rests.  The  negative  electricity  escapes  to  the  ground,  and 
the  disc  is  left  charged  with  positive  electricity,  the  charge  being 
practically  equal  to  that  on  the  resin. 

The  disc  may  now  be  lifted  away,  and  positive  electricity  can  be 
communicated  by  contact  from  it  to  any  conductor ;  and  the  process 
may  be  repeated  from  the  beginning. 

Two  points  in  this  explanation  may  present  some  difficulty.  It 
may  appear  that  the  negative  electricity  of  the  resin  should  be 
destroyed  whenever  the  disc  is  placed  upon  the  surface  of  the  cake  ; 
for  the  induced  positive  electricity  in  the  disc  would  combine  with 
it,  and  then  the  remaining  negative  charge  in  the  disc  would  pass 
to  the  ground  through  the  metallic  connecter.  This  would  really 
happen  if  the  two  surfaces  came  in  contact  throughout  their  whole 
extent ;  but,  because  of  inequalities,  they  only  touch  over  a  com- 
paratively small  area.  Again,  the  negative  electricity  in  the  disc 
is  repelled  by  the  electricity  on  the  resin.  How,  then,  can  it  pass 
to  the  ground  by  a  connection  which  passes  through  the  resin  ? 
We  cannot  be  content  with  the  reply  that  the  disc  and  the  earth 
are  then  both  parts  of  one  conductor,  so  that  a  road  is  opened  up 
by  which  the  electricity  can  get  to  a  greater  distance  from  that 
which  repels  it.  This  statement  seems  very  like  a  statement  to 
the  effect  that  a  stone  would  roll  a  short  distance  up  one  side  of  a 
hill,  in  order  that  it  might  get  a  longer  roll  down  the  other  side. 
The  fact  that  electricity  flows  like  an  incompressible  fluid  (§  335) 
makes  an  explanation  easy.  There  are  three  parallel  layers 
of  electricity  in  the  apparatus — two  negative  layers  with  one 
intermediate  positive  layer.  The  lower  negative  layer  tends  to 
draw  positive  electricity  from  the  ground ;  the  intermediate  positive 
layer  repels  it ;  and  we  are  left  with  the  upper  negative  layer  which 
attracts  it.  We  may  suppose,  therefore,  that  the  negative  layer 
on  the  upper  side  of  the  disc  draws  positive  electricity  from  the 
ground  and  unites  with  it.  The  earth  is  consequently  left  with  an 
equal  negative  charge,  and  the  effect  cannot  be  distinguished  from 


384 


A    MANUAL    OF    PHYSICS. 


that  which  would  have  ensued  npon  an  actual  passage  of  the  negative 
electricity  of  the  disc  to  the  ground.  We  have  no  means  of  dis- 
tinguishing between  the  two  cases,  and  therefore  we  are  justified 
in  saying  that  the  electricity  does  pass  from  the  disc  to  the  ground. 

Returning  from  this  digression,  we  remark  that  the  production 
of  electricity  by  means  of  the  electrophorus  becomes  more  and 
more  continuous  the  more  rapidly  the  various  motions  of  the  disc 
are  performed.  The  principle  of  all  machines  used  for  the  produc- 
tion of  a  statical  charge  is  the  same  as  that  of  the  electrophorus, 
but  they  are  so  constructed  as  to  give  a  strictly  continuous 
production. 

312.  Law  of  Electric  Attraction  and  Repulsion. — We  have 
hitherto  explained  the  various  facts  which  have  come  under  our 
consideration  by  the  assumption  of  attractive  or  repulsive  force, 
which  diminishes  in  intensity  as  the  distance  between  the  attracting 
or  repelling  quantities  increases  ;  and  we  found  that  the  assumption 
enabled  us  to  give  a  consistent  account  of  the  facts.  Therefore, 
adopting  this  assumption  as  a  working  hypothesis,  we  must  now 
consider  more  minutely  the  exact  law  of  force. 

The  law  was  elaborately  investigated  by  Coulomb  by  means  of  his 
torsion  balance.  In  this  instrument  a  vertical  wire  attached  to  the 


FIG.  169. 

torsion-head  h  (Fig.  169)  carries  a  horizontal  insulating  arm,  at  the 
end  of  which  a  small  metal  disc  d  is  fastened. 

A  scale  fastened  to  the  glass  cover  which  surrounds  the  instru- 
ment enables  us  to  determine  the  angular  position  of  the  arm  ;  and 
the  position  of  the  torsion-head,  when  there  is  no  torsion  on  the 
wire,  is  also  noted.  Now  let  a  positive  charge  be  given  to  the  disc 
d,  and  let  another  positive  charge,  contained  in  a  metal  ball  fixed  to 


ELECTROSTATICS.  385 

an  insulating  handle  b,  be  introduced  into  the  interior  through  the 
aperture  a  in  the  glass  cover  of  the  instrument,  the  length  of  the 
handle  being  such  that  the  ball  and  the  disc  are  in  one  horizontal 
plane.  The  mutual  repulsion  of  the  two  quantities  of  electricity  will 
cause  the  arm  to  twist  round  through  a  certain  angle.  Additional 
torsion  is  then  put  on  the  wire,  by  turning  the  head  round,  until  the 
disc  is  brought  back  to  its  former  position. 

Now  increase  the  charge  in  the  ball  in  any  ratio  and  repeat  the 
same  series  of  operations,  having  previously  turned  back  the  torsion- 
head  into  its  old  position.  It  will  be  found  that  the  torsion,  which 
must  now  be  put  on  the  wire  in  order  to  turn  the  disc  back  to  its 
first  position,  is  increased  in  the  same  ratio.  This  proves  that  the 
force  is  proportional  to  the  quantity  of  electricity  in  the  ball,  and 
therefore,  also,  that  it  is  proportional  to  the  quantity  in  the  disc. 

Next,  perform  a  series  of  experiments  in  which  the  charges  of  both 
bodies  are  kept  constant,  while  their  mutual  distance  is  varied. 
The  amount  of  torsion  which  is  requisite  at  the  different  distances 
will  show  that  the  force  varies  inversely  as  the  square  of  the 
distance. 

The  same  law  will  be  found  to  hold  when  the  charges  are  nega- 
tive, and  also  when  one  is  negative  and  the  other  positive.  Of 
course,  in  the  latter  case,  the  angular  rotations  of  the  arm  and  the 
torsion-head  are  necessarily  reversed  in  direction. 

Let  q,  q',  be  the  quantities,  and  let  s  be  the  distance  between 
them.  The  law  of  force  is  expressed  by 


where  the  positive  sign  corresponds  to  repulsion  and  so  indicates 
that  the  force  is  attractive  (i.e.,  is  in  the  direction  of  decreasing  dis- 
tance) when  q  and  q'  are  of  opposite  sign,  and  that  it  is  repulsive 
(i.e.,  is  positive  outwards)  when  they  are  of  like  sign. 

This  is  the  well-known  law  of  gravitational  force,  and,  therefore  all 
results  which  we  have  deduced  (Chap.  VIII.)  regarding  that  force  will 
at  once  apply  to  the  case  of  electric  force,  provided  that  we  take 
account  of  a  possible  reversal  of  sign. 

813.  Electric  Potential.  Electromotive  Force.  —  If  we  attempt  to 
increase  the  charge  of  an  insulated  conductor  by  any  stated  means 
in  which  the  same  conditions  are  maintained,  as  in  the  electro- 
phorus,  we  find  that  it  is  more  and  more  difficult  to  do  so  the  farther 
the  process  is  carried  out,  and  that  the  charge  cannot  be  increased 
beyond  a  certain  limit.  To  make  the  reason  for  this  clear  we  must 
make  an  apparent  digression. 

25 


386  A    MANUAL    OF    PHYSICS. 

Electrified  systems  obviously  possess  energy  in  virtue  of  their 
electrification,  for  the  mutual  attraction  or  repulsion  between  their 
various  parts  may  be  used  for  the  production  of  mechanical  work. 

We  define  the  Mutual  Potential  Energy  of  two  systems  as  the 
amount  of  work  which  may  be  obtained  from  their  mutual  repul- 
sion until  they  are  at  an  infinite  distance  apart ;  and  we  define 
the  Potential  at  any  point,  due  to  a  given  electrical  system,  as  the 
mutual  potential  energy  between  the  system  and  unit  quantity  of 
positive  electricity  placed  at  that  point.  This  definition  makes  the 
sign  of  the  potential  coincide  with  the  sign  of  the  electrification  of 
the  system  to  which  it  is  due,  and  it  makes  the  potential  represent 
potential  energy  and  not  exhaustion  of  potential  energy,  as  in  the 
case  of  gravitation  (§  95). 

Let  V  and  V  be  the  potentials  at  two  points  which  are  at  a  dis- 
tance s  apart.  The  average  force  which  acts  so  as  to  transfer  the  unit 
of  positive  electricity  from  the  point  which  is  at  potential  V  to  the 
point  which  is  at  potential  V  is  (V  -  V)  /s  ;  and,  if  dV  is  the  change 
of  potential  in  the  small  distance  ds  measured  from  any  point,  the 
actual  force  at  that  point  is 

_dV 
ds 

for  our  definition  makes  V  decrease  as  s  increases. 

This  quantity  is  the  rate  of  variation  of  potential  per  unit  length, 
and  is  called  the  Electromotive  Force  at  the  given  point,  for  it  is 
the  force  which  acts  so  as  to  transfer  electricity. 

We  see  therefore  that  no  transference  of  electricity  can  occur 
between  two  conductors  which  are  at  the  same  potential. 

Now  we  have  seen  that  two  like  quantities  of  electricity,  q  and  q', 
situated  at  a  distance  s  apart,  repel  each  other  with  a  force 

q'g, 
~«* 

and  the  force  with  which  q  acts  on  a  unit  of  electricity  of  like 
sign  is 


ds     s'2 

00 


This  gives 


(see  §  35). 

Now,  the  potential  of  a  conductor  must  be  uniform  throughout  if 
the  electricity  which  it  contains  is  at  rest,  for  otherwise  an  electro- 


ELECTROSTATICS.  387 

motive  force  would  act  from  one  part  of  it  to  another.  Hence  we 
are  justified  in  speaking  of  the  potential  of  a  conductor  ;  and  the 
above  expression  shows  that  the  potential  of  a  conductor  becomes 
larger  and  larger  in  strict  proportion  to  the  quantity  of  electricity 
which  it  contains,  being  positive  when  the  charge  is  positive,  and 
negative  when  the  charge  is  negative.  And  it  follows  that  the  work, 
which  must  be  expended  in  order  to  bring  up  to  a  conductor  a  given 
charge,  becomes  greater  and  greater  as  the  charge  already  contained 
in  the  conductor  increases. 

In  particular,  if  the  potential  of  the  body  which  carries  up  the 
new  charge  (the  disc  of  the  electrophorus)  does  not  exceed  a  certain 
fixed  value,  the  potential  of  the  conductor  to  which  the  charge  is 
given  cannot  be  made,  by  this  process,  to  exceed  that  fixed  limit. 
Therefore  the  charge  of  that  conductor  cannot  be  increased  above  a 
fixed  limit,  which  is  the  statement  made  at  the  commencement  of 
this  section. 

The  potential  is  zero  only  at  an  infinite  distance  ;  but,  in  order  to 
practically  carry  a  charge  to  an  infinite  distance,  it  is  necessary 
merely  to  connect  the  conductor,  which  contains  it,  to  the  ground. 
For  the  earth  is  a  conductor  which  is  practically  at  an  infinite  dis- 
tance, and  any  ordinary  charge  which  is  communicated  to  it  pro- 
duces no  sensible  variation  of  its  potential.  (See  next  section.) 

314.  Capacity.  Condensers.  —  The  quantity  of  electricity  which 
must  be  given  to  a  conductor  in  order  that  its  potential  may  be 
raised  by  unity  is  called  the  Capacity  of  that  conductor. 

It  follows  at  once,  from  the  result  of  last  section,  that  the  capaci- 
ties of  two  spherical  conductors  are  in  strict  proportion  to  their 
respective  linear  dimensions.  For,  since  electrostatic  force  obeys 
the  same  law  as  gravitational  force  (§  312)  —  from  which  (§  88)  we 
know  that  the  action  of  each  quantity  is  the  same  as  if  it  were  con- 
densed at  the  centre  so  far  as  points  outside  the  sphere  are  concerned 
—  the  potential  of  a  sphere,  of  radius  a,  is 


Hence,  when  V  =  l,  q  =  a;  that  is,  the  capacity  is  measured  by  the 
radius. 

The  capacity  of  a  compound  conductor  which  consists  of  two 
concentric  spherical  conductors  may  be  made  extremely  great. 

Let  A  and  A'  (Fig.  170)  represent  the  two  surfaces,  and  let  their 
radii  be  respectively  a  and  a+r.  Let  A  be  charged  with  a  quantity 
q  of  positive  electricity,  and  let  A'  be  charged  with  an  equal  quantity 

25—2 


A   MANUAL    OF    PHYSICS. 


of  negative  electricity.     These  two  charges  will  spread  uniformly 
over  the  respective  surfaces  (§  316). 

Now  the  potential  at  any  point  of  A',  due  to  its  own  charge,  is 
;  and  its  potential,  due  to  the  charge  of  A,  is  ql(a  +  T). 


FIG.  170. 

Its  total  potential  is  therefore  zero ;  and  so  no  flow  of  electricity  will 
occur  if  it  be  connected  to  the  ground  by  a  conductor.  This  proves 
that  if  the  sphere  A  be  charged  in  any  way,  an  equal  and  opposite 
charge  will  be  induced  on  A'  if  it  be  connected  to  the  ground. 
Hence,  if  we  add  a  positive  unit  of  electricity  to  A,  a  negative  unit 
will  necessarily  appear  on  A'  when  it  is  *  put  to  earth  ' — to  use  the 
technical  expression.  But  the  work  done  in  increasing  the  charge 
q  of  A  by  unity  is  q/a,  and  the  work  done  in  increasing  the  negative 
charge  —  q  of  A'  by  a  negative  unit  is  -qKa+r).  Hence  the  whole 
'work  done  is 

a     a-\-r      -*•  a(a-\-T) 

When  T  is  very  small,  this  gives  q  =  a2wjr ;  and,  when  w  is  unity, 
q  represents  the  capacity  of  the  arrangement,  which  is  therefore 


a  quantity  which  may  be  made  extremely  great  by  sufficiently 
decreasing  T. 

Any  arrangement  of  this  sort  is  called  a  Condenser,  since  it 
enables  us,  without  much  expenditure  of  work,  to  store  up  a  large 
quantity  of  electricity.  The  only  essential  feature  in  any  such 
arrangement  is  that  the  two  conducting,  and  mutually  insulated, 
surfaces  shall  be  extremely  close  together  in  comparison  with  their 
own  dimensions. 

A  common  form  of  the  condenser  is  known  as  the  Ley  den  jar.  This 
consists  (Fig.  171)  of  a  thin  glass  jar  coated  externally  and  internally 
with  tin-foil.  The  neck  and  upper  portion  of  the  jar  are  not  coated 


ELECTROSTATICS. 


389 


with  the  foil,  so  that  the  insulation  between  the  two  conducting 
sheets  may  be  as  complete  as  possible.  The  mouth  of  the  jar  is 
usually  closed  with  a  cork,  through  which  passes  a  metallic  rod, 
terminating  externally  in  a  knob,  and  making  communication  in- 


FIG.  171. 

ternally  with  the  inner  coating  of  the  jar.  The  external  coating 
can  be  readily  connected  to  the  ground,  while  the  internal  coating 
is  charged,  through  the  agency  of  the  rod,  by  means  of  any  electric 
machine. 

Discharge  of  the  jar  is  effected  by  means  of  the  discharging-rod, 
(Fig.  172),  which  consists  of  two  jointed  metal  rods  connected  to  a 
glass  handle.  The  two  knobs  a  and  6  are  placed  in  connection  with 
the  outer  and  inner  coatings  of  the  jar  ;  and  the  discharge,  resulting 


FIG.  172. 

in  the  combination  of  the  two  equal  and  opposite  quantities  of  elec- 
tricity, takes  place  through  the  metallic  circuit  acb.  Great  care 
must  be  taken  in  the  use  of  jars  charged  to  a  high  potential,  as  very 
serious,  if  not  fatal,  effects  might  ensue  upon  their  discharge 
through  the  human  body. 

Another  common  form  of  condenser  consists  of  a  pile  of  sheets  of 
tin-foil  separated  by  paper  soaked  in  paraffin.  All  the  odd  sheets  in 
the  pile  (counting  from  one  end)  are  connected  together.  So  also 
are  the  even  sheets,  but  the  odd  and  the  even  elements  are  carefully 
kept  separate.  This  arrangement  constitutes  a  condenser  of  great 
capacity. 


390  A  MANUAL   OF   PHYSICS. 

A  practically  identical  arrangement  may  be  made  by  joining 
together  all  the  internal  coatings  and,  independently,  all  the  ex- 
ternal coatings  of  a  number  of  Ley  den  jars.  The  capacity  of  the 
whole  is  equal  to  the  sum  of  the  capacities  of  the  separate  jars. 

If  the  jars  be  connected  together  in  series  —  that  is,  with  the  outer 
coating  of  one  joined  to  the  inner  coating  of  the  next  in  order,  and 
so  on  —  the  resultant  capacity  is  only  equal  to  the  capacity  of  each 
jar  (all  being  supposed  to  be  equal  in  this  respect),  and  the  whole 
charge  is  equal  to  the  charge  of  one  jar  when  its  coatings  are  raised 
to  the  same  (total)  difference  of  potential.  When  the  capacities  are 
not  alike,  we  may  let  V0  and  Vx  indicate  the  potentials  of  the  outside 
and  inside  coatings  of  the  first  jar,  whose  capacity  in  Ca,  while 
Vi  and  V2  are  the  similar  quantities  for  the  second  jar,  the  capacity 
of  which  is  C2,  and  so  on,  V«  being  the  potential  of  the  inner  coat- 
ing of  the  last  jar.  If  C  be  the  total  capacity,  C(Vn-V0)  is  the 
total  charge.  But  this  is  equal  to  the  sum  of  the  separate  charges. 
Hence 


But  the  charge  of  any  one  jar  is  necessarily  equal  to  that  of  any 
other  ;  for  the  outside  coating  of  each  has  a  charge  which  is  equal 
and  opposite  to  that  of  the  inner  coating  of  the  jar  to  which  it  is 
joined,  since  the  two  form  a  single  insulated  conductor  ;  and  the 
charges  in  the  two  coatings  of  any  one  jar  are  also  necessarily  equal. 
Hence  we  have  the  n  —  1  equations 


In  all  there  are  n  equations  connecting  the  2n  +  l  quantities 
C,  d,  .  .  .,  Ca,  V0,  ____  ,  V,.  If  V0=0,  while  the  quantities  d  ---- 
Cn,  are  known,  we  can  eliminate  the  Vs  and  calculate  C. 

315.  Specific  Inductive  Capacity.  —  In  last  section  we  obtained 
the  expression  a2/r  for  the  capacity  of  an  arrangement  consisting  of 
two  concentric  conducting  spherical  surfaces  of  mean  radius  a  and 
separated  by  an  insulating  interval  of  thickness  r. 

It  is  customary  to  consider  air  as  the  standard  insulating  material. 

If  the  air  be  replaced  by  some  other  insulating  material,  such  as 
glass,  resin,  etc.,  it  is  found  that  the  capacity  of  the  same  arrange- 
ment becomes 


where  K  is  a  constant  for  that  particular  medium  and  is  called  its 
Specific  Inductive  Capacity. 


ELECTROSTATICS.  391 

The  Specific  Inductive  Capacity  of  a  substance  may  therefore  be 
determined  by  means  of  measurements  of  the  quantity  of  electricity 
which  is  required  to  charge  a  jar,  of  which  the  given  substance 
forms  the  insulating  medium,  up  to  a  given  potential.  The  ratio  of 
this  quantity  to  the  quantity  which  is  required  in  order  to  raise  the 
potential  of  a  precisely  similar  jar  to  the  same  extent,  when  air  is 
the  insulator,  is  the  value  of  the  required  constant. 

Faraday  determined  the  Specific  Inductive  Capacity  of  various 
substances  by  measurements  of  potential.  He  used  two  precisely 
similar  and  equal  Leyden  jars  which  were  so  constructed  that  the 
insulating  medium  could  be  changed  when  desired.  The  coatings 
of  one  of  the  two  jars  were  insulated  by  air  ;  those  of  the  other  were 
insulated  by  a  substance  whose  (unknown)  inductive  capacity  K, 
was  to  be  found.  He  charged,  the  air  jar  to  potential  V  and  then 
divided  its  charge  between  itself  and  the  other  jar  by  making 
connection  between  their  outer  coatings  and  their  inner  coatings 
respectively.  V  being  the  resultant  common  potential  of  the  jars, 
while  C  is  the  electrostatic  capacity  of  the  air  jar,  the  equation 

V'C+V'KC=VC 

expresses  the  condition  that  the  total  quantity  of  electricity  in  tha 
two  jars  is  equal  to  that  originally  possessed  by  the  air  jar.     Hence 

V-V 


is  found  in  terms  of  the  known  potentials. 

The  value  of  this  constant  (sometimes  called  the  Dielectric  Con* 
stant)  might  also  be  found  by  observations  of  the  reduction  of  the 
potential  of  a  condensing  arrangement,  with  a  given  charge,  when 
the  layer  of  air  between  the  oppositely  electrified  surfaces  is  partially 
displaced  by  a  layer  of  another  insulating  material.  If  t  be  the 
thickness  of  the  new  layer,  the  effective  thickness  of  air  displaced 
is  tjK. 

If  vacuum  be  taken  as  the  standard  insulating  medium,  the 
specific  inductive  capacity  of  air  is  1-00059,  according  to  the  deter- 
minations of  Boltzmann.  Faraday,  with  the  apparatus  at  his  dis- 
posal, was  unable  to  observe  any  difference  between  the  inductive 
capacities  of  different  gases  ;  but  Boltzmann  has  shown  that  small 
differences  really  occur. 

The  dielectric  constants  of  solids  and  liquids  are  greater  than 
those  of  gases  and  differ  considerably  among  themselves. 

316.  Distribution  of  Electricity  on  Conductors.  Electric 
Density.  —  A  static  charge  of  electricity,  which  is  communicated  to 


392  A    MANUAL    OF    PHYSICS. 

a  conductor,  is  necessarily  confined  to  the  surface  of  that  body.  For, 
otherwise,  the  mutual  repulsion  between  like  quantities  of  electricity 
would  produce  continuous  currents  of  electricity  in  the  interior  of 
the  conductor. 

The  fact  that  the  electricity  is  at  rest  on  the  surface  of  the  con- 
ductor, also  shows  that  the  distribution  must  be  such  as  to  produce 
a  uniform  potential  all  over  the  conductor. 

If  the  conductor  be  spherical,  we  see,  from  the  principle  of 
symmetry,  that  the  surface  distribution  of  electricity  is  uniform; 
but  if  the  conductor  be  not  symmetrical,  we  would  infer  that  the 
quantity  of  electricity  distributed,  per  unit  of  area,  over  the  surface 
cannot  be  uniform.  This  quantity  is  called  the  density  of  the 
surface  distribution.  It  is  large  where  the  curvature  of  the  surface 
is  large,  and  is  small  where  the  curvature  is  small. 

In  order  to  determine  the  density  at  various  parts  of  any  surface 
(whether  conducting  or  not),  we  might  place  a  small  plane  metallic 
disc,  attached  to  an  insulating  handle,  in  contact  with  the  surface. 
If  the  curvature  of  the  surface  be  not  too  great,  and  if  the  disc  be 
sufficiently  thin  and  small,  the  electricity  on  the  part  of  the  surface 
covered  by  the  disc  will  be  transferred  to  the  outside  surface  of  the 
disc,  for  it  is  then  practically  a  portion  of  the  electrified  body. 
The  disc  may  then  be  removed,  and  the  magnitude  of  its  charge 
may  be  tested.  If  the  charge  be  q,  while  the  area  is  a,  the 
density  at  that  part  of  the  surface  is  qja.  The  process  may  be 
repeated  as  often  as  we  please,  at  different  portions  of  the  surface, 
BO  long  as  the  total  charge  of  the  conductor  is  not  sensibly 
diminished  ;  and  this  objection  might  be  entirely  avoided  by  taking 
precautions  to  have  the  potential  of  the  conductor  maintained  con- 
stant. As  this  method  was  first  used  by  Coulomb,  the  disc  is  called 
Coulomb's  Proof  Plane. 

In  a  number  of  cases  the  law  of  distribution  of  density  can  be 
calculated  from  the  known  electrostatic  laws.  Some  of  these  cases 
will  be  discussed  in  the  next  two  sections. 

In  particular,  it  is  known  experimentally  that  a  closed  conductor, 
which  has  an  insulated  charge  placed  inside  it,  and  is  connected  to 
the  ground,  will  become  charged  in  such  a  way  that  no  force  is 
exerted  at  any  external  point  by  the  internal  and  superficial  charges ; 
and  the  reason  is  that  its  potential  is  zero,  so  that  the  potential  is 
uniformly  zero  in  all  external  space.  Hence  the  surface  distribu- 
tion, with  its  sign  changed,  acts  at  all  outside  points  precisely 
as  the  internal  charge  does.  Similarly  the  conductor  screens  in- 
ternal bodies  from  the  influence  of  external  charges. 

817.  Electric  Images. — We  have  already  seen  that  a  uniform 


ELECTROSTATICS. 


393 


distribution  of  electricity  over  a  spherical  surface  produces  the  same 
effect  at  external  points  as  an  equal  quantity  of  electricity  con- 
densed at  its  centre  would  produce.  This  imaginary  quantity  is 
called  the  electric  image  of  the  uniform  spherical  distribution. 
[There  is  a  reason  for  calling  this  quantity  imaginary  beyond  the 
mere  fact  that  it  does  not  actually  exist,  and  this  is  that  the  volume 
density  of  a  finite  quantity  of  electricity  condensed  at  a  point  would 
necessarily  be  infinite,  and  such  a  condition  cannot  exist  (§  321). J 
We  proceed  now  to  the  general  discussion  of  the  method  of  electric 
images,  which  is  due  to  Sir  W.  Thomson,  and  which  gives  in  many 
cases  simple  solutions  of  very  formidable  problems. 

Draw  a  sphere  (Fig.  173)  with  radius  CM  =  &,  and  divide  CM  ex- 
ternally and  internally  in  the   points  A,  A'  repectively,   so  that 


FIG.  173. 


CA  .  CA'  =  a2.  Take  any  point  P  in  the  circumference  of  the 
sphere,  and  join  PA,  PA',  PC.  Let  PA  =  r,  PAW,  CA  =  d.  We 
know  by  geometry  that  the  triangles  CAT,  CPA  are  similar. 


Therefore 


CA' 


Now  put 


and  we  get 


CA' 

e'=  -e  — 
a 


394  A   MANUAL   OF   PHYSICS. 

But  if  we  draw  A'Q  perpendicular  to  CA,  we  have  QA  perpen- 
dicular to  CQ,  and  therefore  CA'/a =CA'/CQ  =  CQ/CA=a/d.  Hence 

a 

e'=—  e-. 
d 

If  a  quantity  e  of  electricity  be  placed  at  A,  while  a  quantity  e'  is 
placed  at  A',  the  condition  e'/r' -\-e\r =0  asserts  that  the  potential  at 
the  point  P,  due  to  these  charges,  is  zero  ;  and  therefore  the  sphere 
is  a  surface  of  zero  potential. 

Consequently,  if  the  sphere  be  an  infinitely  thin  conductor,  and 
be  connected  to  the  ground,  the  action  of  the  charge  e  placed  at  A 
will  induce  upon  it  a  charge,  the  effect  of  which  at  external  points 
will  be  the  same  as  that  of  e'  placed  at  A' ;  and  the  charge  e'  placed 
at  A'  will  induce  on  the  sphere  a  charge,  the  effect  of  which  at  in- 
ternal points  is  the  same  as  that  of  e  placed  at  A.  And  further, 
these  two  induced  charges  are  precisely  equal  and  similarly  dis- 
tributed, but  of  opposite  sign  ;  for.  when  the  sphere  is  uncharged  and 
insulated,  and  the  charges  e  and  e'  are  placed  at  A  and  A'  respect- 
ively, the  potential  of  the  sphere  is  zero,  and  no  resultant  charge  is 
induced  upon  it,  neither  can  electricity  flow  from  any  one  part  of  it 
to  any  other  part. 

[To  find  the  law  of  distribution  of  density  which  produces  this 
effect,  produce  AP  to  K,  describe  a  circle  round  the  points  E,  A', 
and  A,  and  let  A'P  meet  this  circle  in  S.  The  triangles  EPS  and 
A'PA  are  similar.  Also  PR  and  PS  are  proportional  to  the  forces, 
/  and  /',  at  P  due  to  e  and  e'  respectively.  For  these  forces  are 
e/r2  and  e'/r'2  respectively,  and  e\r—  -e'/r'.  Hence  the  forces  are 
inversely  proportional  to  r  and  r',  and  therefore  are  directly  pro- 
portional to  PE  and  PS,  by  construction.  Consequently,  the  result- 
ant force  F  at  P  is  proportional  to  SE,  and  so  SE/SP  =  F//'  =  AA'/r. 

AA'       AA'e2  1     AA'e'2  1 

This  gives  F  = /'  =  —    --s  =  —        -7,- 

r  J          e'     r*         e      r'A 

Hence  a  sphere,  the  density  of  which  is  inversely  as  the  cube  of  the 
distance  from  an  external  (or  internal)  point,  attracts  internal  (or 
external)  matter,  as  if  it  were  condensed  at  that  point.] 

The  point  A'  is  the  image  of  A  with  respect  to  the  given  sphere, 
and  by  means  of  the  relation  CA  .  CA'— a2,  we  can  find  the  image 
of  any  distribution  of  electricity. 

For  example,  let  there  be  a  given  distribution  of  electricity  along 
the  straight  line  Aj  A^  (Fig.  174).  The  image  of  the  point  Aj  in  a 
sphere,  of  radius  a,  the  centre  of  which  is  at  C,  is  A'j,  which  is  such 


ELECTROSTATICS.  395 

that   CAi  .  CAf!  =  a2.      Similarly,  CA'2=A'2=a2,  and   so  on.     It  is 
easy  to  prove  that  the  line  A'iA'a  is  a  portion  of  circle.     But  if 


FIG.  174. 

AiA2  is  very  small  A^A'g  is  practically  straight,  and  is  inclined  to 
AiC  at  an  angle  equal  to  the  angle  of  inclination  between  CjA 
and  AjAo. 

Let  CAj  =  r,  CA'^r',  A1Aa=Z,  A'jA'^Z',  and  we  get,  as  the 
ratio  of  an  infinitely  small  length  in  the  image  to  an  infinitely  small 
length  in  the  direct  system, 


Similarly,  «'  and  a  representing  areas,  v'  and  v  representing 
volumes, 

^_^!_r'4.  £!_^L_?!l! 

«      r4     a,4'    v     r6     a6 

Next  let  X,  <r,  p,  represent  line,  surface,  and  volume  densities  in  the 
direct  system,  and  let  X',  a',  p'  be  the  corresponding  densities  in  the 
image.  We  get 


_^_e_o.  r__a  _r 
\  ~  ej~  e  l'~~  r  r'~  r'     a' 
1 

Similarly,  ^=e-  °=  a-  '£  =  *=*  ;£=*=£. 

a      e  a!      r  a4     a3     r«    p     a5     r's 

Again,  if  V  and  V  represent  direct  and  image  potentials,  we  get 


r  =ar  _  a  _r 
r'~r  r'~  r'~  a 


It  is  upon  this  relation,  in  terms  of  which  we  can  find  the  distribu- 
tion of  potential  in  the  inverted  system  when  we  know  the  distribu- 
tion of  potential  in  the  direct  system,  that  the  physical  use  of  the 
method  of  electrical  images  essentially  depends. 


396  A    MANUAL    OF    PHYSICS. 

For  example,  let  a  sphere  be  charged  uniformly  with  electricity, 
and  let  us  invert  it  with  respect  to  itself.  In  that  case,  while  the 
image  coincides  with  the  direct  system,  each  point  outside  the  image 
corresponds  to  a  point  inside  the  original  sphere.  But 


and  we  know  that  V  is  constant.  Hence  the  potential  at  a  point 
outside  a  uniformly  charged  sphere  is  inversely  proportional  to  the 
distance  of  that  point  from  its  centre. 

Next,  invert  the  uniformly  charged  sphere  with  respect  to  an 
external  point.  The  inverted  system  is  also  a  sphere,  and  the 
density  of  the  distribution  upon  it  is  given  by 

a* 

0  =  °~7* 
r'3 

where  a  is  constant.  Hence  the  density  of  the  image  sphere  varies 
inversely  as  the  distance  of  the  centre  from  the  point  of  inversion. 
Further,  points  inside  the  first  sphere  invert  into  points  inside  the 
image.  Hence  the  potential  equation  shows  that  the  potential  at  an 
internal  point  varies  inversely  as  the  distance  of  that  point  from  the 
centre  of  inversion,  so  that  internal  material  points  are  attracted  as 
if  the  sphere  were  condensed  at  the  centre  of  inversion. 

Finally,  invert  the  uniformly  charged  sphere  with  respect  to  an 
internal  point.  The  density  of  the  image  sphere  varies  inversely  as 
the  cube  of  the  distance  from  the  centre  of  inversion,  and  the 
potential  varies  inversely  as  that  distance,  but  internal  points  have 
inverted  into  external  points.  Hence  external  material  points  are- 
attracted  by  a  sphere,  whose  density  varies  inversely  as  the  cube  of 
the  distance  from  an  internal  point,  as  if  the  sphere  were  condensed 
at  that  point. 

These  three  propositions  have  already  been  otherwise  proved.  A 
comparison  of  the  present  proofs  with  those  previously  given  will 
exhibit  to  a  slight  extent  the  great  power  and  simplicity  of  the 
method  of  electric  images. 

318.  Electric  Lines  of  Force.  —  As  we  have  already  seen,  all  the 
results  obtained  in  Chap.  VIII.  regarding  gravitational  potential 
and  force  may  be  at  once  applied  to  the  treatment  of  electrical 
potential  and  force.  We  may  surround  a  given  electrostatic  system 
by  equipotential  surfaces,  and  we  may  suppose  lines  of  force  to  be 
drawn,  perpendicular  to  the  equipotential  surfaces,  in  such  a  way 
that  the  number  of  lines  drawn  outwards,  per  unit  area,  from  such 
a  surface  represents  the  electric  force  at  that  part  of  the  surface. 


ELECTROSTATICS.  397 

The  results  of  that  chapter  then  enable  us  to  state  that  the  density 
of  electricity  at  any  point  of  a  charged  surface  is  l/4?r  times  the 
number  of  lines  of  force  which  originate  (or  end)  per  unit  area,  on 
the  surface  at  that  point,  the  density  being  positive  or  negative 
according  as  the  lines  originate  or  end  at  the  surface,  that  is, 
according  as  they  are  drawn  outwards  from,  or  inwards  to,  it. 
Similarly,  the  positive  (or  negative)  volume  density  of  electricity  at 
any  point  of  space  is  l/4?r  times  the  number  of  lines  which  originate 
(or  end)  per  unit  of  volume  at  that  point. 

Since  no  line  of  force  can  originate  or  end  except  at  a  point  where 
electricity  is  situated,  and  since  experiment  shows  that  no  electrical 
effect  is  felt  outside  a  closed  conductor,  connected  to  the  ground, 
which  completely  surrounds  any  electrical  system,  we  see  that  all 
the  lines  of  force  which  originate  at  the  parts  of  the  system  must 
end  (except  in  so  far  as  they  may  proceed  from  one  part  of  the 
system  to  another)  upon  the  surrounding  conductor ;  and,  con- 
sequently, the  charge  induced  on  the  conductor  is  equal  in  amount 
and  opposite  in  sign  to  the  total  algebraic  sum  of  the  various 
internal  charges.  A  particular  case  of  this  was  discussed  in 
§  314. 

Every  line  proceeds  from  a  point  at  high  potential  to  a  point  at 
low  potential.  It  therefore  originates  on  a  positively  charged  body 
and  ends  on  a  negatively  charged  body  or  passes  to  infinity. 

319.  Electric  Induction.  Tubes  of  Induction. — Hitherto  we  have 
spoken  of  gravitational  and  electrical  action  as  if  it  took  place 
directly  at  a  distance.  We  have  spoken  of  the  mutual  potential 
energy  of  two  systems  without  inquiring  how  one  system  possesses 
energy  in  virtue  of  its  position  relatively  to  another.  But,  if  we 
believe  that  energy  is  transferred  by  means  of  matter  (§  7),  we  must 
look  upon  some  intervening  medium  as  the  vehicle  through  which 
it  is  transferred. 

This  is  the  way  in  which  Faraday  regarded  the  subject ;  and  most 
of  the  development  of  electrical  science  in  recent  times  is  due  to 
Faraday's  work  together  with  Clerk- Maxwell's  mathematical  inter- 
pretation and  development  of  his  views.  The  fact,  stated  in  last 
section,  that  the  electrical  condition  of  charged  conductors  depends 
upon  the  nature  of  the  intervening  insulating  medium  gives  strong 
support  to  this  belief. 

In  Chap.  XXIX.  we  shall  see  that,  when  a  current  of  electricity 
flows  along  a  conductor,  the  amount  of  electricity  which  crosses  any 
section  of  the  conductor,  per  unit  of  time,  is  constant.  In  other 
words,  the  flow  of  electricity  resembles  the  flow  of  an  incompressible 
fluid.  Similarly,  the  facts  that  no  quantity  of  one  kind  of  electricity 


398  A    MANUAL    OF    PHYSICS. 

can  appear  without  the  simultaneous  development  of  an  equal 
quantity  of  the  opposite  kind,  and  that  the  quantity  of  electricity 
induced  on  a  closed  conductor,  which  entirely  surrounds  the  inducing 
electricity,  is  equal  in  quantity,  and  opposite  in  sign,  to  the  latter, 
indicate  that  the  induction  of  electricity  through  a  dielectric  re- 
sembles the  displacement  of  an  incompressible  fluid.  [This  fact, 
probably,  prolonged  the  use  of  the  objectionable  term  '  electric  fluid.'] 

We  must  therefore  look  upon  a  conductor  as  a  body  which  cannot 
sustain  electrostatic  stress,  and  so  permits  electricity  to  flow  along 
it  when  such  stress  is  applied ;  and  we  are  to  regard  an  insulator  as 
a  body  which  can  sustain  electrostatic  stress,  in  which  '  displace- 
ment '  of  electricity  takes  place  in  proportion  to  the  stress  which  is 
applied,  and  in  which  the  displacement  is  annulled  when  the  stress 
is  removed.  In  this  way  of  looking  at  the  matter,  a  surface  charge 
is  supposed  to  reside  on  the  surface  of  the  dielectric  and  not  on  the 
surface  of  a  conductor. 

The  word  '  displacement '  was  introduced  by  Maxwell  from  the 
analogy  to  an  elastic  medium  the  parts  of  which  suffer  displacement 
when  stress  is  applied,  and  recover  from  the  distortion  when  the 
stress  is  removed.  But  Maxwell  was  very  careful  to  avoid  attaching 
any  exact  meaning  to  the  term  '  electric  displacement.'  He  merely 
used  it  by  analogy  ;  and  this  cannot  be  too  carefully  kept  in  view. 

Faraday  used  the  term  Electric  Induction  to  indicate  the  state  of 
the  medium  in  virtue  of  which  equal  and  opposite  quantities  of 
electricity  appear  on  opposed  surfaces ;  and  Maxwell  spoke  of  the 
total  amount  of  induction  through  a  given  surface  as  the  amount  of 
electricity  which  is  '  displaced '  through  it.  A  forward  displacement 
in  an  insulator  corresponds  to  a  direct  current  in  a  conductor ;  a 
diminution  of  displacement  corresponds  to  a  reverse  current. 

The  difference  between  the  specific  inductive  capacities  of  various 
substances  is  explained  by  a  difference  in  the  amount  of  displace- 
ment which  is  produced  in  each  under  the  same  electromotive 
force. 

We  may  draw  lines  of  force  through  all  points  of  any  small  closed 
curve  on  a  conductor  so  as  to  form  a  tube  of  force ;  and  we  may 
draw  such  tubes,  covering  the  whole  surface  of  the  conductor,  in 
such  a  way  that  the  number  emanating  per  unit  of  area  from  all 
parts  of  the  surface  is  equal  to  4™  when  <j  is  the  density  of  the 
electrification.  The  number  of  such  tubes,  which  intersect  unit 
area  of  any  equipotential  surface,  therefore  expresses  the  intensity 
of  the  force  at  that  part  of  the  equipotential  surface. 

But,  instead  of  proceeding  in  the  above  manner,  we  may  draw 
the  tubes  so  that  each  encloses  unit  amount  of  electrification  on  the 


ELECTROSTATICS.  399 

conductor.  Faraday  called  such  tubes  Tubes  of  Induction ;  for, 
when  they  originate,  they  enclose  unit  quantity  of  positive  electricity, 
and,  when  they  end,  they  enclose  unit  quantity  of  negative  electricity. 
The  total  number  of  tubes  of  induction  originating  from,  or  ending 
on,  a  conductor,  expresses  its  total  positive,  or  negative,  electrifica- 
tion, and  the  '  induction,'  or  '  displacement,'  through  each  section  of 
such  a  tube  is  constant. 

Properly  speaking,  tubes  of  induction  are  formed  by  lines  of 
induction,  and  not  by  lines  of  force.  And  it  is  well  to  remember 
that  tubes  of  induction  are  not  necessarily  tubes  of  force :  for  the 
displacement  does  not  always  take  place  in  the  direction  of  the 
electromotive  force,  although  in  general  it  does.  (Compare  the 
elastic  properties  of  non-isotropic  solids,  §  245.) 

320.  Electric  Energy. — In  order  to  estimate  the  amount  of 
energy  which  is  associated  with  the  charge  of  a  conductor  at  a  given 
potential,  we  have  merely  to  calculate  the  work  expended  in  charg- 
ing it.  Let  Q  be  the  charge,  and  let  V  be  the  potential.  Let  us 
suppose  that  the  conductor  is  charged  by  successive  infinitesimal 
instalments  dq,  and  that  the  charge  at  any  instant  is  q,  while'  the 
capacity  of  the  conductor  is  C.  The  potential  at  the  given  instant 
is  therefore  q/C.  But  the  potential  is  the  work  which  is  required  in 
order  to  bring  up  unit  charge  from  an  infinite  distance,  and  give  it 
to  the  conductor.  Hence  the  amount  of  work  which  is  necessary  in 
order  to  increase  the  charge  q  by  the  amount  dq  is 

dq_ 
C 

and  the  total  amount  of  work  which  musi  be  expended  in  raising 
the  charge  from  0  to  Q  is 

0 
C  /J        Q2 

•i~c~~~ac"~~ 

Hence  the  whole  energy  is  one  half  of  the  product  of  the  charge 
into  the  potential ;  or  one  half  of  the  product  of  the  capacity  into 
the  square  of  the  potential ;  or  one  half  of  the  quotient  of  the  square 
of  the  charge  by  the  capacity. 

Now  let  us  look  at  the  problem  from  the  point  of  view  of  induc- 
tion. Consider  a  positively  electrified  body  inside  a  closed  con- 
ductor. Draw  unit  tubes  of  induction  from  the  body  to  the  internal 
surface  of  the  conductor,  and  describe  equipotential  surfaces  corre- 
sponding to  all  potentials  which  differ  from  each  other  by  unity, 
and  are  included  between  the  potential  V  of  the  electrified  body  and 


400  A   MANUAL    OF    PHYSICS. 

the  potential  V  of  the  surrounding  conductor.  The  number  of  cells 
into  which  the  tubes  of  induction  and  the  equipotential  surfaces 
divide  the  volume  between  the  two  charged  surfaces  is  equal  to  the 
product  of  V  —  V  into  the  number  of  tubes,  i.e.,  into  the  charge  of 
the  body.  Hence  the  number  of  cells  into  which  the  space  is 
divided  is  double  of  the  electrical  energy  of  the  system. 

A  simple  extension  of  this  reasoning  shows  that  the  same  result 
is  true  whatever  be  the  number  of  electrified  bodies  contained  inside 
the  conductor.  (See  Maxwell's  Elementary  Treatise  on  Electricity, 
Chap.  V.) 

This  result  points  to  the  conclusion  that  the  energy  of  a  system  of 
charged  conductors  is  contained,  not  in  the  conductors  themselves, 
but  in  the  insulating  medium  which  surrounds  them.  And  Fara- 
day's and  Maxwell's  views  of  the  nature  of  induction  show  us  how 
this  may  be.  The  dielectric  is  in  a  state  of  strain  so  long  as  dis- 
placement is  maintained  by  the  action  of  electromotive  force  ;  so 
that  the  energy  which  was  expended  in  producing  the  strain  is  con- 
tained in  the  dielectric  in  a  potential  form.  [Visible  strain  may  be 
produced  in  a  piece  of  glass  by  means  of  electrostatic  stress.] 

To  obtain  an  expression  for  the  amount  of  energy  contained,  per 
unit  of  volume,  in  the  dielectric,  let  us  consider,  as  the  simplest 
case,  an  insulated  sphere  of  variable  radius  r  charged  with  a  con- 
stant quantity  q  of  positive  electricity.  The  potential  of  the  sphere 
is  qjr,  and  the  energy  contained  in  the  space  external  to  it  is 


Now  let  r  increase  infinitesimally  to  r-\-dr.     The  energy  becomes 

2?  • 
The  difference  of  these  quantities, 


is  the  energy  which  is  contained  in  the  intervening  shell  of  volume 
47rr-dr.  Hence  the  energy  contained  per  unit  of  volume  at  the  dis- 
tance r  is 

E-ilSj.lp. 

STT  r4      STT 
where  F  is  the  resultant  force  at  the  distance  r  due  to  the  charge  q. 


ELECTROSTATICS.  401 

If  K  be  the  specific  inductive  capacity  of  the  medium  we  must 
write 

E  =  —  F». 

STT 

This  result  is  quite  general,  F  being  the  resultant  force  at  any  point 
due  to  the  total  electrification. 

Maxwell  has  investigated  the  nature  of  the  stress  in  the  dielectric 
which  would  account  for  observed  electric  phenomena.  He  finds 
that  the  stress  consists  of  a  tension  KF2/47r  along  the  lines  of  force 
coupled  with  an  equal  pressure  in  all  directions  at  right  angles  to 
the  lines  of  force. 

In  particular,  the  tension  at  the  charged  surface  of  a  conductor  is 
47rK<r2  in  a  direction  perpendicular  to  the  surface,  where  a  is  the 
surface  density  of  the  electrification. 

As  an  instructive  example  in  connection  with  the  above  expres- 
sion for  the  energy  of  an  electrified  system,  we  may  estimate  the 
energy  of  a  charge  Q,  first  when  it  is  contained  in  a  jar  of  capacity 
C,  and,  second,  when  it  is  divided  between  that  jar  and  another  jar 
of  capacity  C'.  The  original  energy  is 


After  division,  since  the  potentials  of  the  two  jars  are  equal,  we 
have 


= 

C      C'' 

where  Q3  and  Q.2  are  the  charges  in  the  jars  of  capacities  C  and  C 
respectively.     Also 

Qi+Q2=Q, 

and  hence  QlS=Q_5_,  Q^Q^SL. 

And  the  respective  energies  are 


Therefore,  the  total  energy  is 


which  is   always    less  than   the    original   energy.      The   fraction 
C'/(C+Cf)  of  the  whole  energy  has  been  dissipated  in  the  process 

26 


*-Q— 

2n  j^nr' 


402  A   MANUAL    OF   PHYSICS. 

of  division — usually  taking  the  form  of  sound,  light,  and  heat.  [Com  - 
pare  the  dissipation  of  energy,  which  takes  place  when  a  gas  is 
allowed  to  expand  without  doing  work,  as  in  Joule's  experiment 
(§  303),  or  that  which  occurs  when  heat  diffuses,  so  as  to  arrive  at 
a  lower  temperature  without  the  performance  of  work.  In  the 
present  case  no  electricity  is  lost,  but  the  potential  is  lowered.] 

321.  Electric  Absorption.  Disruptive  Discharge. — If  an  elastic 
medium  be  distorted  beyond  its  limits  of  perfect  elasticity,  the 
removal  of  the  stress  is  not  followed  by  complete  recovery  from 
strain ;  but  if  the  distortion  be  not  too  great,  complete  recovery 
may  take  place  after  a  sufficient  time  has  elapsed.  Conversely,  a 
long-continued  force  may  produce  large  distortion. 

Analogous  phenomena  appear  in  dielectric  media  when  subjected 
to  electrostatic  stress. 

Thus,  a  Leyden  jar,  when  charged  to  a  certain  potential,  will 
gradually  fall  in  potential,  though  it  is  well  insulated.  The  result 
is  the  same  as  if  its  capacity  gradually  increased,  or  as  if  the  specific 
inductive  capacity  gradually  increased  so  that  the  same  displace- 
ment was  maintained  by  a  smaller  difference  of  potential.  If  the 
jar  be  now  discharged,  the  quantity  of  electricity  which  is  obtained 
is  smaller  than  the  original  charge.  This  phenomenon  is  known  as 
Electric  Absorption,  for  the  jar  appears  to  have  absorbed  some  of 
its  charge. 

A  second  (small)  discharge  may  be  obtained  from  the  jar  if  it 
be  left  for  some  time.  Subsequently  a  third  may  be  obtained,  and 
so  on  until  the  total  discharge  equals  the  original  charge.  These 
are  called  Eesidual  Discharges.  The  apparently  absorbed  charge 
seems  to  leak  out  again,  as  if  the  medium  gradually  recovered 
from  a  temporary  distortion. 

It  is  not  well  to  pursue  these  analogies  too  far.  Maxwell  has 
shown  that  apparent  absorption  will  take  place  if  the,  insulating 
medium  is  heterogeneous  in  the  sense  that  it  consists  of  parts,  the 
specific  inductive  capacities  of  which  differ  from  one  another,  or  of 
parts  which  differ  from  each  other  in  their  insulating  power. 
The  insulating  power  of  ordinary  dielectrics,  such  as  glass,  gutta 
percha,  etc.,  is  not  perfect ;  and  so,  if  a  composite  dielectric  con- 
sisted of  alternate  layers  of  incompletely  and  completely  insulating 
materials,  electric  absorption  would  be  manifested. 

We  know  also  that  elastic  solids  are  only  capable  of  withstanding 
strain  to  a  limited  extent,  and  that  they  will  be  ruptured  if  too 
great  stress  be  applied.  Similarly  all  dielectrics  will  cease  to  insu- 
late electricity  if  they  are  subjected  to  too  great  electrostatic  stress. 
The  state  of  strain  in  the  insulating  material  of  a  Leyden  jar 


ELECTROSTATICS.  403 

becomes  greater  and  greater  as  the  potential  of  the  jar  is  raised 
higher ;  but,  if  the  process  be  continued  too  far,  the  insulation 
breaks  down,  and  the  separated  electricities  recombine  through  the 
ruptured  dielectric.  This  phenomenon  is  called  the  Disruptive 
Discharge. 

A  Leyden  jar,  through  the  substance  of  which  the  disruptive  dis- 
charge has  occurred,  is  useless  for  all  subsequent  electrical  purposes, 
for  the  glass  is  in  part  shattered  by  the  discharge.  On  the  other 
hand,  if  air  or  any  other  fluid  were  used  as  the  dielectric,  the  jar 
would  insulate  as  completely  as  ever  it  did  so  long  as  too  great 
stress  were  not  again  applied ;  for,  although  by  the  energy  of  the 
discharge  the  parts  of  the  fluid  medium  would  be  violently  dis- 
rupted, the  insulation  would  be  restored  by  an  inflow  of  the  sur- 
rounding medium. 

The  disruptive  discharge  is  usually  accompanied  by  the  produc- 
tion of  sound,  light,  heat,  and  mechanical  effect,  the  total  energy 
evolved  being  the  exact  equivalent  of  the  original  electrical  energy. 

Various  forms  of  the  disruptive  discharge  exist.  The  most 
ordinary  form  is  called  the  spark  discharge.  When  two  oppositely 
charged  surfaces  are  brought  sufficiently  near  each  other,  the  electro- 
static stress  in  the  medium  increases  to  such  an  extent  that  the 
electricities  combine  because  of  rupture  of  the  dielectric  between 
the  charged  surfaces.  A  small  streak  of  light  is  apparent  where 
the  discharge  occurs,  its  form  depending  upon  the  thickness  of  the 
dielectric  through  which  the  discharge  occurs.  When  the  distance 
is  great  the  streak  of  light  (the  spark)  is  very  irregular  and  jagged 
in  outline. 

Feddersen  found  that  the  nature  of  the  spark  discharge  depends 
upon  the  resistance  (§  335)  of  the  circuit  in  which  the  discharge 
occurs.  When  the  resistance  is  sufficiently  large,  it  consists  of 
successive  rapid  discharges  in  the  same  direction.  It  becomes 
continuous  when  the  resistance  is  lessened  to  a  certain  extent,  and 
it  consists  of  a  rapidly  alternating  series  of  discharges  in  opposite 
directions,  when  the  resistance  is  still  further  diminished. 

Sometimes  the  discharge  is  in  the  form  of  a  brush.  This  is  seen 
chiefly  when  one  of  the  two  conductors  has  great  curvature  at  the 
place  where  the  discharge  occurs.  A  short  line  of  light,  which 
abruptly  branches  out  into  a  brush-like  form,  appears  at  the  place. 
Wheatstone  showed,  by  means  of  his  revolving  mirror,  that  the  brush 
discharge  consists  of  a  series  of  rapidly  succeeding  separate  dis- 
charges. Its  intermittent  character  gives  rise  to  the  crackling,  or 
even  musical,  sound  which  accompanies  this  form  of  the  dis- 
charge. 

26—2 


404  A    MANUAL    OF    PHYSICS. 

The  glow  discharge  takes  place  from  the  rounded  extremity  of  a 
wire  which  projects  into  the  air.  The  end  of  the  wire  is  covered  by 
a  phosphorescent  light.  This  form  of  the  discharge  does  not  appear 
to  be  intermittent.  It  seems  rather  to  be,  as  Faraday  concluded,  a 
convective  discharge,  in  which  the  charge  is  carried  away  by  the 
particles  of  the  air.  (Compare  the  action  of  the  pith-ball,  §  307.) 

The  '  electric  wind,'  which  blows  from  a  sharp  electrified  point,  is 
due  to  the  repulsion  of  air  particles  which  have  been  electrified  by 
contact  with  the  point. 

The  limiting  tension  (§  320)  which  the  insulating  medium  can 
sustain  without  rupture  is  called  the  Dielectric  Strength  of  the 
medium.  The  dielectric  strength  of  air  depends  upon  the  distance 
between  the  oppositely  electrified  surfaces.  It  has  a  greater  value 
when  the  distance  is  small  than  it  has  when  the  distance  is  large. 
In  all  gases  it  increases  as  the  pressure  increases,  and  diminishes  as 
the  pressure  diminishes — but  not  indefinitely.  A  minimum  value 
is  reached  at  a  certain  stage,  beyond  which  the  strength  increases 
as  the  pressure  is  farther  diminished. 

The  method  of  spark  discharge  under  diminished  pressure  (in 
so-called  vacuum  tubes)  is  much  used  for  the  purpose  of  examina- 
tion of  the  spectra  of  gases. 

322.  Atmospheric  Electricity,  etc. — The  atmosphere  is  almost 
always  in  a  state  of  electrification,  either  positive  or  negative.  The 
electrification  is  generally  positive  during  long- continued  fine 
weather ;  it  generally  becomes  negative  when  the  fine  weather  breaks. 

In  order  to  test  the  nature  of  the  electrification,  use  may  be  made  of 
Thomson's  water-dropping  accumulator.  This  instrument  consists 
of  an  insulated  metallic  vessel,  which  contains  water,  and  which  is 
fitted  with  a  long  fine  nozzle,  from  which  the  water  issues  drop  by 
drop  when  the  stopcock  with  which  it  is  fitted  is  opened.  The 
nozzle  projects  out  into  the  external  atmosphere  by  an  opening  in 
the  window,  and  the  vessel  is  connected  with  an  electrometer.  The 
stopcock  is  then  opened  and  the  water  drops  out. 

If  the  atmosphere  be  positively  electrified,  negative  electricity 
will  be  induced  in  the  nozzle  and,  therefore,  in  the  drop,  while 
positive  electricity  is  repelled  to  the  electrometer.  As  each  drop 
falls  away,  carrying  its  negative  charge  with  it,  the  vessel  and 
electrometer  are  left  more  and  more  positively  charged.  The  electri- 
fication of  the  atmosphere  is,  therefore,  indicated  by  the  development 
of  a  charge  of  like  sign  in  the  electrometer. 

An  interesting  question  arises  in  this  connection— What  is  the 
source  of  the  energy  of  the  charge  in  the  electrometer?  The 
energy  of  the  charge  may  be  transformed  into  heat,  and  the  energy 


ELECTROSTATICS.  405 

of  the  falling  drops  may  also  be  transformed  into  heat.  Further, 
there  is  no  other  possible  source  of  heat  in  the  arrangement.  But 
the  drops  may  fall  without  any  production  of  electric  charge,  and, 
therefore,  the  principle  of  conservation  compels  us  to  assert  that 
the  drops  will  fall  more  slowly  when  they  are  electrified  than  they 
do  when  unelectrified,  and  so  will  do  less  work.  This  conclusion  is 
verified  by  experiment. 

If  we  replace  the  metallic  vessel  (above  alluded  to)  by  a  hot 
crucible,  into  which  we  drop  water,  the  water  will  evaporate,  and 
the  vapour  will  be  found  to  be  negatively  electrified,  for  the  crucible 
and  the  electrometer  become  positively  charged.  If  the  vapour 
condenses,  the  total  volume  of  all  the  drops  of  water  which  are 
formed  remains  constant;  but  the  total  surface  of  the  drops 
diminishes  as  each  drop  increases  in  size,  and  the  same  quantity  of 
electricity  is  confined  to  a  smaller  surface.  The  result  is  that  the 
potential  of  the  drops  rises  considerably.  It  is  possible  that  the 
high  potential  of  thunder-clouds  may  be  explained  in  this  way. 

When  the  potential  rises  to  such  an  extent  that  the  air  is  unable 
to  withstand  the  electrostatic  stress,  disruptive  discharge  (lightning) 
takes  place. 

The  great  use  of  a  lightning-rod  is  to  prevent  the  potential  from 
rising  to  such  an  extent  that  disruptive  discharge  will  occur.  It 
does  this  by  drawing  off  from  the  surrounding  air  a  continuous 
current  of  electricity.  The  electrified  air  induces  the  opposite 
electrification  in  the  rod,  and  the  density  is  very  great  at  the  sharp 
point — so  great,  that  the  electricity  streams  off  from  it  to  the  air  by 
silent  discharge,  and  so  annuls,  totally  or  partially,  the  electrifica- 
tion of  the  air ;  and  this  is  equivalent  to  the  passage  of  the  opposite 
electricity  from  the  air  to  the  ground  through  the  rod.  If  a  cloud 
in  the  neighbourhood  of  the  rod  were  suddenly  electrified  to  a  high 
potential  by  disruptive  discharge  from  a  distant  thunder-cloud,  the 
rod  may  not  be  able  to  draw  off  the  electricity  with  a  sufficient 
rapidity  to  prevent  discharge  from  the  near  cloud  to  the  building 
supposed  to  be  protected  by  the  rod.  The  rod  would,  more  likely 
than  not,  be  insufficient  for  the  purpose  of  carrying  off  the  discharge 
to  the  ground. 

323.  Pyroelectricity. — If  a  crystal  of  tourmaline,  or  of  some  other 
minerals,  be  heated,  electrical  phenomena  will  be  manifested, 
although  previously  the  crystal  appeared  to  be  unelectrified.  Posi- 
tive electricity  appears  at  one  end  of  the  crystallographic  axis, 
negative  electricity  appears  at  the  other. 

The  electrification  may  now  be  destroyed  by  passing  the  crystal 
through  a  flame.  Further  electrification  will  then  be  manifested, 


406  A   MANUAL    OF    PHYSICS. 

similar  to  that  which  formerly  appeared,  if  the  heating  be  pro- 
ceeded with  ;  but  if,  on  the  contrary,  the  crystal  be  allowed  to  cool, 
.the  opposite  electrifications  will  appear  at  the  ends. 

Sir  W.  Thomson  supposes  that  such  crystals  possess  internal 
electrification — that  they  are  electrically  polarised  in  the  direction 
of  the  axis — and  that,  when  they  are  passed  through  the  flame, 
their  surfaces  become  electrified  in  such  a  way  as  to  annul  at  all 
external  points  the  effect  of  the  internal  electrification.  And  he 
further  supposes  that  the  amount  of  internal  electrification  depends 
upon  the  temperature,  so  that  heating  or  cooling  disturbs  the 
balance  of  external  effects. 

324.  Electrification  by  Contact. — The  electrification  of  glass  or 
of  sealing-wax,  etc.,  by  friction  may  be  explained  by  the  assumption 
that  an  electromotive  force  exists  at  the  surface  of  contact  of  the 
two  substances  which  tends  to  produce  electric  displacement  across 
the  interface,  and  that  friction  is  used  merely  for  the  purpose  of 
securing  better  contact. 

The  substances  being  non-conductors,  the  electricity  cannot  pass 
from  the  surface,  and  the  displacement  continues  until  the  effect 
of  the  reverse  force  which  it  entails  balances  the  effect  of  the 
electromotive  force  of  contact.  So  long  as  the  surfaces  remain 
in  contact,  the  arrangement  acts  as  a  condenser  of  extremely  large 
capacity,  and  relatively  large  displacement  may  be  produced  by  a 
comparatively  feeble  difference  of  potential.  But  whenever  the 
surfaces  are  separated,  the  potential  rises  greatly,  because  of  the 
ensuing  decrease  of  capacity.  In  this  way  the  high  potential  obtain- 
able from  frictional  machines,  or  from  the  electrophorus,  etc.,  is 
explained.  The  charge  is.  the  same  after  separation  as  before  it,  but 
the  potential  has  increased,  and  therefore  the  energy  has  increased  ; 
and  the  increase  of  energy  is  the  precise  equivalent  of  the  work  done 
in  the  process  of  separation. 

Now,  although  we  cannot  electrify  conductors  by  friction  in  the 
,way  that  we  electrify  non-conductors,  we  can  produce  electrification 
of  conductors  by  contact  or  friction,  provided  that  we  take  proper 
precautions. 

If  we  take  two  flat  pieces  of  zinc  and  copper,  insulate  them  both, 
and  then  place  their  flat  faces  in  contact,  the  copper  will  become 
negatively  electrified,  while  the  zinc  becomes  positively  electrified. 
This  may  be  proved  by  separating  the  plates  (still  insulated)  and 
testing  their  electrification  by  means  of  the  electroscope  or  the 
electrometer.  And  we  may  explain  the  result  by  stating  that  an 
electromotive  force  of  contact  acts  at  the  surface  of  separation  of 
the  metals  in  the  direction  from  copper  to  zinc. 


ELECTROSTATICS.  407 

Volta  found  that  this  assumed  electromotive  force  of  contact 
between  any  pair  of  metals  is  equal  to  the  sum  of  the  electromotive 
forces  between  every  pair  of  metals  forming  a  series  closed  by  the 
given  pair.  From  this  it  would  follow  that  the  sum  of  the  contact 
forces  in  any  complete  heterogeneous  metallic  circuit  is  zero.  This 
is  known  to  be  true  so  long  as  the  temperature  is  uniform  through- 
out the  circuit ;  and  it  is  in  accordance  with  the  principle  of  conser- 
vation of  energy,  for  there  is  no  source  of  energy  in  such  a  circuit. 

The  assumption  of  the  existence  of  an  electromotive  force  of 
contact  between  metals  sufficiently  great  to  account  for  the 
observed  effects  is  regarded  as  inadmissible  by  many  physicists. 
That  a  contact  force  does  exist  is  shown  by  thermoelectric  pheno- 
mena ;  but  this  force  is  very  much  smaller  than  the  Volta  contact 
force.  Consequently,  those  physicists  who  are  unwilling  to  admit 
the  possibility  of  a  true  contact  force  between  metals,  which  would 
account  for  the  whole  observed  effect,  look  upon  the  surfaces  of 
separation  between  the  metals  and  the  air  as  the  real  seat  of  the 
electromotive  force.  The  whole  question  is  still  involved  in  much 
uncertainty. 

Contact  forces  exist  between  metals  and  liquids,  and  also  between 
different  liquids.  Volta's  law  does  not  hold  universally  in  the  latter 
case. 

325.  The  Electrometer. — Instruments  such  as  the  gold-leaf  elec- 
troscope may  be  used  for  the  purpose  of  obtaining  very  rough 
measurements  of  electromotive  force.  The  electrometer,  in  one  or 
other  of  its  various  forms,  is  used  when  accurate  measurements  of 
electrostatic  effects  are  required.  It  is  used  directly  for  the  deter- 
mination of  difference  of  potential  and  it  may  be  indirectly  used 
for  the  purpose  of  the  comparison  of  the  capacities  of  conductors, 
and  consequently  for  the  determination  of  their  charges. 

Instruments  such  as  the  gold-leaf  electroscope  are  termed  idiostatic 
instruments,  for  there  is  no  electrification  in  any  part  of  these  instru- 
ments except  such  as  is  due  to  the  electrification  which  is  to  be  tested ; 
and  their  indications  (when  small)  are  therefore  proportional  to  the 
square  of  the  difference  of  potential  which  is  to  be  observed.  Any 
small  variation  of  potential  is  therefore  inappreciable  when  the 
potential  itself  is  small,  and  the  indications  of  the  instrument  are 
the  same  kind,  whether  the  potential  is  positive  or  negative.  In 
heterostatic  instruments,  some  of  which  we  shall  now  describe,  a 
constant  charge  of  one  definite  kind  is  maintained  in  one  part  of  the 
apparatus,  so  that  a  small  variation  of  potential  produces  the  same 
effect,  whether  the  potential  is  large  or  small,  and  the  indication  is 
reversed  in  direction  when  the  potential  changes  sign. 


408  A   MANUAL   OF   PHYSICS. 

Most  forms  of  the  electrometer  depend  for  their  action  upon  the 
electrostatic  force  between  similarly  or  oppositely  charged  bodies. 

Coulomb's  torsion  balance  is  therefore  one  (but  a  very  imperfect) 
form  of  electrometer. 

In  the  attracted  disc  electrometer  the  two  charged  bodies  are  in 
the  form  of  parallel  horizontal  discs  placed  at  a  distance  apart 
which  is  small  in  comparison  with  their  transverse  dimensions. 
We  shall  assume  that  the  discs  are  oppositely  charged,  the  densities 
of  the  electrifications  being  -f  <r  and  —  o-  respectively.  Except  in  the 
near  neighbourhood  of  the  edges,  the  lines  of  force  are  perpendicular 
to  the  discs,  and  the  force  at  any  point  between  them  is  (§  99) 
27nr--27r(—  <7)=47nr  in  the  direction  from  the  positively  charged  disc 
to  the  negatively  charged  disc.  Since  %-n-a  is  the  force  with  which 
the  positively  charged  disc  acts  upon  unit  quantity  of  negative 
electrification  on  the  other  disc,  the  total  force  with  which  it  attracts 
that  disc  is  27r<r.ou  =  27r0-2<x,  where  a  is  the  area  of  the  disc.  But, 
as  above,  the  force  at  any  point  between  the  discs  is  47r<r,  and  is 
equal  to  (V-V')/£,  where  V  and  V  are  respectively  the  potentials 
of  the  positively  and  the  negatively  charged  discs,  and  t  is  the 
interval  between  them.  Hence  a=  (V-  V')/47r£,  and  the  total  force 
of  attraction  between  the  discs  is 

1  (V-V)2 
=&—^*> 

which  gives  V  -  V  =  t  A  /—  F. 

V      a 

[We  might  have  deduced  this  result  from  the  expression  for  the 
energy  contained  in  unit  volume  of  the  dielectric  (§  320),  which  is 


STT          8?r       t2 

Hence  the  total  energy  contained  in  the  volume  at  between  the 
discs  is 

1      (V-V)2 

—  -—   ,    —  —  —  —    —     •   Ct* 

STT  t 

And  the  rate  at  which  this  varies  per  unit  of  thickness  gives  the 
force 


In  Thomson's  absolute  electrometer,  in  which  this  principle  is  used, 
a  concentric  circular  portion  of  the  upper  disc  is  alone  moveable,  and 
so  the  difficulty  of  the  non-uniformity  of  the  force  at  the  edge  of  the 


ELECTROSTATICS. 


409 


discs  is  avoided.  The  moveable  part  as  nearly  as  possible  fills  the 
aperture  without  touching  its  sides,  and  it  is  suspended  by  means  of 
a  delicate  (spring)  balance,  which  has  a  fiducial  mark  by  means  of 
which  the  lower  face  of  the  movable  disc  can  always  be  placed  in 
one  plane  with  the  lower  face  of  the  surrounding  portion  of  the 
upper  disc — called  the  guard-ring.  The  lower  disc  can  be  accurately 
moved,  perpendicular  to  its  own  plane,  through  known  distances, 
by  means  of  a  screw.  The  balance  and  disc  are  surrounded  by  a 
metal  case  for  the  purpose  of  preventing  any  disturbance  which 
might  arise  from  external  electrification.  When  the  two  discs  are 
connected  to  bodies  of  different  potential,  the  balance  is  depressed, 
and  the  screw  is  turned  until  it  returns  to  its  standard  position.  In 
this  way  the  distance  t  is  determined.  Also,  by  previous  experi- 
ments, it  is  known  what  weight  must  be  placed  on  the  disc  in  order 
to  bring  it  to  its  standard  position,  and  this  gives  the  (constant) 
value  of  F. 

In  using  the  instrument  it  is  preferable  to  keep  its  lower  disc  at  a 
constant  potential  by  means  of  a  charged  condenser — the  constancy 
being  determined  by  means  of  a  secondary  electrometer.  The  value 
of  t  is  then  found,  first,  when  the  upper  disc  is  connected  to  the 
ground,  and  again,  when  it  is  connected  to  the  body  whose  potential 
is  to  be  determined.  If  9  be  the  difference  of  the  distances,  the 
above  equation  gives  us  the  value  of  the  potential 


-„      /87TF 

-°v  -^-' 


a  being  now  the  area  of  the  attracted  disc. 

In  Thomson's  quadrant  electrometer  an  aluminium  needle  swings 
inside  a  hollow  metal  cylinder,  which  is  divided  into  four  quadrants. 


FIG.  175. 

The  two  opposite  pairs  of  quadrants  (Fig.  175)  are  connected  by 
wires.     The  needle  in  its  normal  position  is   suspended  with 


410 


A    MANUAL    OF    PHYSICS. 


length  directed  along  one  of  the  lines  of  division  between  the 
quadrants,  and  it  is  charged  to  a  high  positive  potential.  One  pair 
of  quadrants,  say  those  connected  to  the  wire  a,  may  be  connected 
to  the  ground,  while  the  other  pair  is  connected  by  the  wire  6  to  a 
body  at  positive  potential  V.  The  quadrants  connected  with  b 
become  positively  charged,  and  the  quadrants  connected  with  a 
become  negatively  charged.  The  needle  is  therefore  deflected 
towards  the  negatively  charged  quadrants,  and  the  deflecting  couple 
is  proportional  to  V,  if  the  potential  of  the  needle  be  sufficiently  high. 

Modifications  of  this  instrument  may  be  used  for  the  measure- 
ment of  extremely  small,  and  of  extremely  large,  differences  of 
potential. 

326.  Electric  Machines.  —  The  electrophorus,  which  is  the 
simplest  form  of  electrical  machine,  has  been  already  described. 

As  an  example  of  the  older  machines,  used  for  the  continuous 
production  of  electricity,  we  shall  take  the  cylinder  machine.  This 
machine  consists  of  a  glass  cylinder  C  (Fig.  176),  which  is  turned 


FIG.  176. 

round  in  the  direction  AmB.  An  insulated  leather  rubber  A,  coated 
with  zinc  amalgam,  presses  against  the  rotating  cylinder,  and  causes 
the  development  of  positive  electricity  on  the  glass,  while  it  becomes 
itself  negatively  electrified.  The  positive  electricity  is  carried  round 
on  the  surface  of  the  glass  until  it  reaches  the  sharp  metal  points  p, 
which  project  from  the  insulated  metallic  conductor  B.  It  induces 
in  the  points  negative  electricity  which  is  discharged  on  the  surface 
of  the  glass,  destroying  its  electrification,  and  leaving  B  positively 
electrified.  A  silk  flap  m,  connected  to  the  rubber  and  resting  in 
contact  with  the  upper  portion  of  the  cylinder,  prevents  the  electri- 
fication from  slipping  back  along  the  surface  of  the  glass.  The 
potential  of  the  positive  electricity  rises  rapidly  as  it  is  carried 
from  A  to  B,  and  the  resulting  electromotive  force  may  cause  the 


ELECTROSTATICS. 


411 


electricity  to  slip  back,  so  that  the  potential  of  B  cannot  rise  very 
high.  The  silk  flap  becomes  negatively  electrified  and  so  prevents 
the  slipping,  or  stops  it  if  it  does  occur  by  using  the  slipping  elec- 
tricity to  destroy  its  own  negative  electrification,  which  tends  to 
equalise  the  potential. 

In  the  plate  machine  the  glass  cylinder  is  replaced  by  a  circular 
glass  plate,  which  is  rubbed  on  both  sides. 

The  Holtz  machine  is  one  of  the  best  modern  forms  of  electric 
machines.  A  fixed  glass  disc  D  (Fig.  177)  has  two  paper  armatures 
(a,  a'}  fixed  on  it  near  the  opposite  extremities  of  a  diameter.  Near 
each  armature  an  opening  (shown  by  dotted  lines)  is  cut  in  the 
glass,  through  which  a  paper  point  attached  to  the  armature  pro- 
jects so  as  nearly  to  come  in  contact  with  a  revolving  glass  disc  C, 


JL 


FIG.  177. 

which  is  mounted  upon  an  axis  passing  through  the  centre  of  D. 
A  metal  conductor  ra,  fitted  with  a  row  of  sharp  points,  faces  the 
revolving  disc  on  the  other  side  opposite  the  armature  a.  A  similar 
conductor  n  faces  the  armature  a',  and  can  be  placed  in  communi- 
cation with  m  by  means  of  the  rod  Z,  which  slides  through  the 
knob  m. 

In  order  to  work  the  machine,  the  knobs  n  and  m  are  placed  in 
connection,  and  a  charge  (positive,  say)  is  given  to  the  armature  a 
by  means  of  the  electrophorus  or  otherwise,  while  the  disc  is  rotated 
in  the  direction  aCa'.  After  a  short  time  a  rustling  sound  is  heard, 
and  the  machine  becomes  difficult  to  drive.  It  is  producing  elec- 


412  A    MANUAL    OF    PHYSICS. 

tricity,  and  the  extra  work  which  has  to  be  performed  is  the  equiva- 
lent of  the  electrical  energy  which  is  developed. 

We  may  explain  the  process  of  charging  in  the  following  manner  : 
The  positive  charge  given  to  a  induces  negative  electrification  in  the 
points  of  the  conductor  m.  This  is  discharged  upon  the  glass  sur- 
face, and  the  glass  carries  it  round  to  the  opposite  side  of  the 
machine,  leaving  m  positively  charged.  Here  it  induces  positive 
electricity  in  the  sharp  point  of  the  armature  a',  and  this  is  dis- 
charged upon  the  inside  of  the  glass  disc,  leaving  a'  negatively 
charged.  The  armature  a'  now  draws  positive  electricity  to  the 
points  of  the  conductor  n.  This  electricity  is  discharged  upon  the 
outer  surface  of  the  disc,  leaving  the  conductor  n  negatively  charged. 
Thus  we  may  regard  the  glass  disc  as  constantly  carrying  positive 
electricity  from  n  to  m  in  the  one  half  of  its  revolution,  and  as 
constantly  carrying  negative  electricity  from  m  to  n  in  the  other 
half  of  its  revolution. 

It  is  possible  that  the  negative  charge  is  given  to  a'  by  way  of  the 
conductor  n.  The  positive  electricity,  which  is  produced  in  m  by 
the  first  motion  of  the  machine,  and  flows  to  n,  may  be  supposed  to 
act  inductively  on  the  armature  a',  drawing  negative  electricity  to 
the  body  of  it,  and  repelling  positive  electricity  to  the  sharp  point 
to  be  discharged  upon  the  glass,  leaving  a'  negatively  charged. 

When  the  conductors  n  and  m  are  slightly  separated,  a  brush 
discharge  passes  through  the  air  space.  This  brush  discharge  may 
bo  changed  into  a  violent  spark  discharge  by  connecting  the  inner 
coatings  of  two  Leyden  jars  to  the  conductors  n  and  m,  the  outer 
coatings  of  the  jars  being  joined  together.  The  jars  have  to  be 
charged  up  to  the  potential  required  for  discharge  through  the  air 
space,  and,  as  their  capacities  are  large,  a  great  quantity  of  elec- 
tricity passes  at  each  discharge. 


CHAPTEE    XXVIII. 

THERMO-ELECTRICITY. 

327.  Thermo-electric  Phenomena. — Though  the  principle  of  con- 
servation of  energy  shows  (§  324)  that  the  sum  of  the  electromotive 
force  in  a  closed  metallic  circuit  must  be  zero,  provided  that  there 
be  no  difference  of  temperature  between  the  various  parts  of  the 
circuit,  we  cannot  assert  that  their  sum  will  be  zero  if  the  tempera- 
ture be  not  uniform.  For,  in  the  process  of  equalisation  of  tempera- 
ture, it  is  possible  that  there  may  be  transformation  of  thermal 
energy  into  electric  energy ;  and  this  transformation  will  occur  if 
the  electromotive  forces  of  contact  between  the  metals  which  form 
the  circuit  depend  upon  the  temperature. 

Now,  Seebeck  discovered  in  1822  that,  in  general,  a  current  of 
electricity  flows  around  a  circuit,  which  is  composed  of  two  different 
metals,  if  there  be  a  difference  of  temperature  between  the  two 
junctions  of  the  metals ;  and  this  shows  that  the  equilibrium  of  the 
contact  forces  has  been  destroyed  because  of  the  variation  of  tem- 
perature. 

And  we  cannot  assert,  without  experimental  evidence,  that  there 
will  be  no  resultant  electromotive  force  in  a  closed  circuit  composed 
of  a  single  metal  which  varies  in  temperature  from  point  to 
point.  But  the  experiments  of  Magnus  showed  that  no  such  force 
exists. 

Still,  in  order  that  Magnus's  result  should  hold,  it  is  necessary 
that  every  part  of  the  circuit  should  be  physically  similar  and  not 
merely  chemically  similar.  For  example,  two  portions  of  the  same 
metallic  substance,  which  are  in  a  different  state  of  strain,  are 
physically  different  substances,  and  are  also  thermo- electrically 
distinct.  And  two  portions  of  the  same  substance  which  are  at 
finitely  different  temperatures  are  in  different  physical  states,  and 
might,  therefore,  exhibit  thermo-electric  phenomena.  Such  phe- 
nomena were  observed  by  Le  Roux  and  others  at  the  instant  when 


414  A   MANUAL   OF   PHYSICS. 

contact  was  made  between  two  portions  of  the  same  metal  which 
differed  abruptly  in  temperature. 

328.  Laws  of  Thermo-electric  Circuits. — It  is  found  by  experi- 
ment that  the  introduction  of  a  piece  of  metal  into  a  thermo- 
electric circuit  does  not  contribute  to  the  electromotive  force  of 
that  circuit,  provided  that  the  extremities  of  the  metal  are  at  one 
and  the  same  temperature.  We  'may,  therefore,  use  solder  to 
connect  together  the  various  parts  of  the  circuit. 

Let  the  lines  A  (Fig.  178)  represent  two  pieces  of  the  same 
metal.  Let  two  of  their  ends  have  a  common  temperature  £1}  while 
the  other  two  have  a  common  temperature  ta ;  and  let  the  ends 
which  are  at  the  temperature  ^  be  joined  by  a  metal  C,  while  the 
ends  which  are  at  the  temperature  ta  are  joined  by  a  metal  B.  As 
the  whole  arrangement  is  symmetrical  with  respect  to  the  pieces  A, 


FIG.  178. 

it  is  obvious  that  there  can  be  no  resultant  electromotive  force  in 
the  circuit.  And  if  the  temperature  of  one  of  the  junctions  between 
C  and  A  be  changed  from  ^  to  t.2,  the  metal  B  will  still,  from  its 
symmetrical  position,  contribute  nothing  to  the  electromotive  force, 
although  there  may  now  be  a  resultant  electromotive  force  in  the 
circuit. 

Next  let  us  arrange  pieces  of  two  metals  alternately,  as  in 
Fig.  179,  and  let  the  temperatures  of  their  extremities  be  as  indi- 
cated. The  pieces  B,  which  are  at  temperatures  ^  and  t.2  respect- 
ively, contribute  nothing  to  the  total  effect,  so  that  the  whole 
arrangement  really  consists  of  two  metals  (A  and  B),  the  two 
junctions  of  which  are  t3  and  t0  respectively.  Now  we  may  join 
the  pieces  ^B^  and  £2B£2  respectively  to  those  points  of  the  piece 
t.jBt01  which  are  at  the  temperatures  t±  and  t.2,  by  means  of  con- 
nections made  of  the  metal  B  ;  and  these  new  connections  contribute 
nothing  to  the  total  electromotive  force  in  the  circuit.  But  the 
electromotive  force  in  the  part  BB^A^B  is  due  to  the  metals  B  and 
A  when  the  temperatures  of  the  junctions  are  ^  and  t0.  Similarly, 
the  force  in  the  part  BB^A&jB  is  due  to  A  and  B,  with  temperatures 
t:  and  t.2  at  the  junctions ;  and  that  in  the  part  BB£2A£3  is  due  to  A 


THERMO-ELECTRICITY. 


415 


and  B  with  temperature  t.2  and  t.A  at  the  junctions.  Hence  the  elec- 
tromotive force,  due  to  temperatures  t.A  and  t0,  is  equal  to  the  alge- 
braic sum  of  the  electromotive  forces  due  to  temperatures  t3  and 
t0,  t.2  and  ti,  £3  and  t.2,  respectively. 

fc         B 


FIG.  179. 

This  result  is~quite  general,  and,  therefore,  the  algebraic  sum  of 
the  various  electromotive  forces  in  a  compound  circuit,  which  is 
composed'^of  a  number  of  pieces  of  two  metals  ivith  their  junctions 
at  various  temperatures,  t0  and  £1?  t±  and  &>, . .  . .  tn-\  and  tm  is  equal 
to  the  electromotive  force  in  a  tivo  functioned  circuit  of  the  same 
metals  with  its  junctions  at  the  extreme  temperatures  t0  and  tn. 

We  can  thus  obtain  a  comparatively  large  electromotive  force  by 
means  of  a  small'  difference  of  temperature.  This  is  the  essential 
principle  of  the  Thermopile,  an  instrument  which,  in  conjunction 
with  a  galvanometer,  is  used  for  the  measurement  of  small  differ- 
ences of  temperature. 

Lastly,  arrange  four  metal  wires,  A,  B,  C,  and  B  in  the  manner 


FIG.  180. 

indicated  in  Fig.  180,  and  let  the  one  wire  B  be  raised  to  tempera- 
ture t»  while  the  other  is  kept  at  temperature  t0.  The  wires  B 
merely  serve  as  junctions,  and  so  the  total  electromotive  force  is  due 
to  a  circuit  composed  of  A  and  C  with  the  junctions  at  tempera- 
tures h  and  t0  respectively.  But  we  may  join  the  two  wires  B  by 
third  wire  of  the  same  metal  without  altering  the  distribution  of 
electromotive  force  in  the  circuit.  And  in  the  circuit  B£jA£0BB  the 


416 


A   MANUAL   OF  PHYSICS. 


electromotive  force  is  due  to  A  and  B  with  junctions  at  temperatures 
£1  and  t0 ;  while  in  the  circuit  B^C^BB  the  electromotive  force 
is  due  to  B  and  C,  with  junctions  at  the  same  temperatures. 

If  we  define  the  thermo-electric  power  of  a  circuit  of  two  metals 
as  the  rate  at  which  the  electromotive  force  in  that  circuit  varies 
per  unit  of  difference  in  temperature  between  the  junctions,  we  see 
that  the  result  which  we  have  just  obtained  shows  that  at  every 
temperature  the  thermo-electric  power  of  A  and  C  is  the  algebraic 
sum  of  the  thermo-electric  powers  of  A.  and  B,  and  B  and  C. 

329.  Variation  of  the  Electromotive  Force  with  Temperature. — 
Soon  after  Seebeck's  discovery  Gumming  observed  that  in  certain 
circuits  (such  as  that  of  iron  and  copper),  while  one  junction  is  main- 
tained at  a  constant  ordinary  temperature  and  the  other  is  gradually 
raised  in  temperature,  the  electromotive  force  gradually  increases 
to  a  maximum,  then  diminishes,  vanishes,  and  finally  is  reversed. 

The  law  of  variation  of  the  electromotive  force  has  been  very  fully 
investigated  by  Gaugain  and  others.  They  found  that,  with  most 
pairs  of  metals,  the  curve  which  is  obtained  by  plotting  difference 
of  temperature  along  the  (horizontal)  axis  of  a?,  and  electromotive 


force  along  the  (vertical)  axis  of  y  (Fig.  181)  is  in  general  a  parabola 
with  its  axis  vertical.  Therefore,  if  we  denote  the  electromotive 
force  by  e,  and  the  temperature  by  t  and  let  E  and  T  respectively 
represent  the  electromotive  force  and  the  temperature  which  cor- 
respond co  the  vertex  of  the  parabola,  we  obtain 

E-e  =  &(T-£)2 (1) 

where  6  is  a  constant ;  for  this  equation  merely  expresses  the  well- 
known  property  of  the  parabola,  that  the  square  of  the  distance  of 


THERMO-ELECTRICITY. 


417 


any  point  on  it  from  the  axis  is  proportional  to  the  distance  of  that 
point  from  the  tangent  at  the  vertex. 

In  particular  cases  the  curve  is  a  straight  line ;  in  others,  it  is 
made  up  of  portions  of  parabolas  with  parallel  (vertical)  axes,  but 
with  their  vertices  alternately  turned  in  opposite  directions. 

330.  The  Thermo-electric  Diagram. — Now  another  well-known 
property  of  the  parabola  is  that  the  rate  of  increase  of  the  ordinate 
at  any  point  per  unit  of  increase  of  the  abscissa  is  proportional  to 
the  value  of  the  abscissa  measured  from  the  axis.  The  proof  of  this 
is  simple,  for  (1)  gives 


(2), 


which  is  the  direct  expression  of  the  above  statement. 

If,  therefore,  we  plot  the  value  of  dejdt  (which  is  the  thermo- 
electric power)  against  difference  of  temperature,  we  shall  obtain, 
instead  of  a  parabola,  a  straight  line.  And  we  inay  form  a  self- 
consistent  diagram  of  such  lines  for  any  number  of  metals  by  means 
of  observations  on  circuits  consisting  of  each  of  these,  in  turn,  with 
some  standard  metal  whose  line  is  made  to  coincide  with  the  axis  of 
temperature.  (In  the  true  diagram  the  lines  might,  of  course,  be 


T  T!      T2 

FIG.  182. 

curves  obtained  from  these  straight  lines  by  a  process  of  shearing.) 
We  see,  by  (2),  that  these  various  lines  will  intersect  the  axis  of 
temperature  at  points  which  correspond  to  the  temperatures  of 
the  maximum  ordinates  in  the  original  diagram  (Fig.  181).  And, 
further,  the  point  of  intersection  of  any  pair  of  lines  in  the  diagram, 
indicates  the  temperature,  T,  at  which  the  electromotive  force  in  a 
circuit  of  the  two  corresponding  metals  attains  a  maximum  value. 
This  diagram  is  called  the  thermo-electric  diagram. 

Sir  W.  Thomson  first  suggested  the  construction  of  such  a  diagram. 
The  actual  construction  of  it,  upon  the  assumption  (suggested  by 

27 


418  A  MANUAL   OF   PHYSICS. 

theoretical  considerations)  that  the  curves  which  represent  the  thermo- 
electric powers  of  the  metals  are,  in  general,  straight  lines,  is  due  to 
Tait.  The  diagram  on  page  422  is  reduced  from  his  results. 

The  area  included  between  two  temperature  lines  and  the  lines 
which  represent  the  thermo-electric  position  of  any  pair  of  metals 
represents  the  electromotive  force  in  a  circuit  of  these  metals  when 
the  junctions  are  kept  at  the  two  given  temperatures.  This  follows 
(see  §  34)  from  the  way  in  which  the  diagram  of  lines  has  been 
deduced  from  the  diagram  of  parabolas. 

331.  The  Peltier  Effect. — A  thermo-electric  circuit  forms  a  system 
which  is  in  stable  equilibrium.  For,  if  it  were  in  unstable  equi- 
librium, increase  of  temperature  of  one  of  the  junctions  would 
produce  effects  which  would  still  further  increase  the  temperature. 
But  we  know  that  the  application  of  heat  to  one  junction  produces 
a  current  of  electricity  which  flows  in  a  certain  direction  across  that 
junction.  Therefore  by  the  principle  of  stable  equilibrium  (§  15), 
we  can  assert  that  the  passage  of  electricity  in  the  given  direction 
will  cool  the  junction. 

The  current  at  the  hot  junction  always  flows  from  the  metal 
which  has  the  lower  thermo-electric  power  to  the  metal  which  has 
the  higher  thermo-electric  power.  Conversely,  heat  is  absorbed  at  a 
junction  where  a  current  flows  in  this  direction,  or  is  evolved  at  a 
junction  where  the  current  flows  in  the  reverse  direction.  Peltier 
discovered  this  by  direct  experiments  made  without  reference  to  any 
theoretical  considerations ;  and  so  the  phenomenon  of  the  absorption 
or  disengagement  of  heat  at  a  junction  across  which  electricity  flows 
is  known  as  the  Peltier  Effect  at  that  junction.  The  total  Peltier 
effect  in  any  circuit  vanishes  when  the  two  junctions  are  at  the 
same  temperature,  for  the  absorption  of  heat  at  one  of  the  junctions 
is  equal  to  the  disengagement  of  heat  at  the  other. 

S32.  The  Thomson  Effect. — In  order  to  explain  the  fact  that,  in 
auch  circuits  as  iron-copper,  the  direction  of  the  electromotive  force 
changes  when  the  hot  junction  is  sufficiently  raised  in  temperature, 
Thomson  assumed  that  the  Peltier  effect  vanishes  at  that  tempera- 
ture at  which  the  electromotive  force  reaches  its  maximum  value, 
that  is,  at  the  temperature  at  which  the  lines  of  the  metals  intersect 
in  the  thermo-electric  diagram.  The  metals  are  then  said  to  be 
neutral  to  each  other,  and  so  this  temperature  is  called  the  Neutral 
Temperature. 

Now  no  heat  is  being  absorbed  or  developed  at  the  junction  which 
is  at  the  neutral  temperature,  and  heat  is  being  developed  at  the 
cold  junction,  for  there  the  current  is  flowing  from  the  metal  of 
higher  thermo-electric  power  to  the  metal  of  lower  thermo-electric 


THERMO-ELECTRICITY.  419 

power.  It  would  seem,  therefore,  that  there  is  no  source  of  thermal 
energy  in  the  circuit  by  the  transformation  of  which  we  can  account 
for  the  development  of  electrical  energy.  But  there  is  no  other 
possible  source  of  the  electrical  energy,  and  hence  Thomson  was  led 
to  predict  that  heat  is  absorbed  at  parts  of  the  circuit  other  than  the 
junctions,  either  in  that  metal  in  which  the  current  flows  from  hot 
parts  to  cold  parts,  or  in  that  metal  in  which  it  flows  from  cold  parts 
to  hot  parts,  or  in  both  metals.  And  he  subsequently  verified  his 
prediction  by  direct  experiment. 

In  copper,  heat  is  absorbed  when  the  current,  is  passing  from  cold 
parts  to  hot  parts  ;  in  iron,  it  is  absorbed  when  the  current  is  passing 
from  hot  parts  to  cold  parts.  [It  is  assumed  here  that  we  know  the 
direction  in  which  a  current  is  flowing.  The  convention  by  which 
this  is  determined  will  be  stated  in  next  chapter.]  Now,  in  a  tube 
through  which  a  liquid  is  flowing,  heat  is  absorbed  by  the  liquid 
when  it  passes  from  cold  parts  to  hot  parts.  Hence,  in  copper  and 
similar  metals,  electricity  acts  as  an  ordinary  fluid  would  do  ;  and  so 
Thomson  speaks  of  the  specific  heat  of  electricity.  It  is  positive  in 
copper  and  similar  metals,  and  is  negative  (at  ordinary  temperatures, 
at  least)  in  iron  and  similar  metals. 

333.  Further  Discussion  of  the  Thermo-electric  Diagram.  —  The 
Peltier  and  the  Thomson  effects  can  be  readily  represented  on  the 
thermo-electric  diagram. 

Let  el  represent  the  electromotive  force  in  a  certain  circuit  gom- 
posed  of  some  metal  together  with  the  standard  one.  T,  being  the 
neutral  temperature,  we  get,  at  temperatures  t  and  t'  respectively, 
while  the  temperature  t0  of  the  cold  junction  remains  constant 


whence 


Similarly,  if  we  use  any  other  metal  giving  the  electromotive  force 
e.2  with  the  standard  metal  under  the  same  conditions 

. .  (4). 

Hence  the  electromotive  force  in  a  circuit  of  the  two  given  metals 
under  the  same  conditions  as  to  temperature  is 


27—2 


420 


A   MANUAL   OF   PHYSICS. 


Now  (2)  shows  us  that  26X  and  262  are  the  tangents  of  the  angles  at 
which  the  lines  of  the  metals  meet  the  line  of  the  standard  metal  in 


the  thermo-electric  diagram.  So,  if  we  let  t0  represent  the  absolute 
zero  of  temperature,  while  all  the  other  temperatures  are  given  in 
absolute  units,  we  get 

qt0  =  26^!  and  pt0  =  262T2 

and  so  (Fig.  183)  qp  =  2(61T1  -  62T2).     But  qp  =  2T(6X  -  &2), 
whence  b^  -  62T2  =  Tfa  -62), 


-  -~,  ....  (5), 


and  therefore    tf-=2(61- 


which  is  the  general  expression  for  the  electromotive  force  in   a 
circuit  of  two  metals  in  terms  of  the  inclinations  of  their  lines  to  the 
line  of  the  standard  metal,  of  their  neutral  temperature,  and  of  the 
temperatures  of  the  junctions. 
Now  (5)  may  be  put  in  the  form 

^=2(61-62)(T-e')(^-^)  +  (&i-&2)(^'-e)2  ....  (6) 

The  first  term  on  the  right-hand  side  of  (6)  vanishes  when  t'  =  t  and 
also  when  the  temperature  of  the  hot  junction  is  at  the  neutral 
point.  It  therefore  represents  the  part  of  the  electromotive  force 
which  corresponds  to  the  Peltier  effect.  Therefore,  if  there  is  no 
other  effect  than  the  Peltier  effect  and  the  Thomson  effect,  the 
second  term  must  represent  the  part  of  the  electromotive  force  which 
corresponds  to  the  Thomson  effect. 

If  we  suppose  that  unit  quantity  of  electricity  is  transferred  along 
the  circuit  under  the  electromotive  force  <?,  the  quantity  e  represents 


THERMO-ELECTRICITY.  421 

the  electric  energy  which  is  expended  in  the  process,  and  therefore 
the  quantity  on  the  right-hand  side  of  the  equation  expresses,  on  the 
one  hand,  the  heat  which  is  absorbed  in  the  production  of  the  electric 
energy,  or,  on  the  other  hand,  the  heat  which  is  evolved  when  unit 
quantity  of  electricity  passes  round  the  circuit  under  the  electro- 
motive force  e.  The  first  term  of  (6)  is  therefore  taken  as  the 
measure  of  the  Peltier  effect,  while  the  second  term  is  taken  as  the 
measure  of  the  Thomson  effect. 

If  the  current,  flows  round  the  circuit  in  the  direction  abcda,  the 
area  abcda  represents  the  whole  heat  which  is  absorbed  in  the 
circuit  during  the  passage  of  unit  quantity  of  electricity. 

But  2(61  -  62)  (T—  t'}—ab,  and  therefore  the  area  abnma  represents 
the  heat  which  is  on  the  whole  absorbed  at  the  junctions.  The 
whole  area  abrsa  represents  the  heat  which  is  absorbed  at  the  hot 
junction,  for  at  that  junction  the  current  is  passing  from  the  metal 
of  lower  thermo-electric  power  to  the  metal  of  higher  thermo-electric 
power ;  while  the  area  msrnm  represents  the  heat  which  is  evolved 
at  the  cold  junction,  for  at  the  cold  junction  the  current  passes  from 
the  metal  of  high  thermo-electric  power  to  the  metal  of  low  thermo- 
electric power. 

Again,  cn  =  e2,b^(t'—  t),  and  therefore  the  triangular  area  cnb  = 
%  .  2&!(£'—  t]  (f  -  t)  represents  the  heat  which  is  absorbed  in  the  metal 
of  higher  thermo-electric  power.  Similarly  the  area  amd  represents 
the  heat  which  is  evolved  in  the  metal  of  lower  thermo-electric 
power.  [The  former  part  has  the  positive  sign  prefixed  to  it  in 
(6)  ;  the  latter  part  has  the  negative  sign  prefixed.] 

It  is  evident  that  we  may  state  quite  generally  that  heat  is 
absorbed  at  any  part  of  the  circuit  at  which  the  current  is  passing 
from  parts  of  lower  thermo-electric  power  to  parts  of  higher  thermo- 
electric power,  and  is  evolved  at  any  part  when  the  current  is  pass- 
ing from  parts  of  higher  to  parts  of  lower  thermo-electric  power. 

The  term  (6X  —  &2)  (V  -  t)  (V  -  t)  may  be  regarded  as  the  product  of 
the  sum  of  the  average  specific  heats  of  electricity  in  the  various  parts 
of  the  circuit  into  the  range  of  temperature,  each  term  in  the  sum 
being  positive  or  negative  according  as  it  corresponds  to  absorption 
or  evolution  of  heat.  Therefore  the  form  of  the  various  terms  b^t', 
bj,  bst't  and  bst,  shows  that  the  specific  heat  of  electricity  in  any 
metal  is  proportional  to  the  absolute  temperature,  and  consequently 
the  average  which  must  be  taken  is  the  arithmetical  mean ;  so  that, 
if  a  be  the  actual  specific  heat  at  temperature  £,  the  average  value 
throughout  the  range  t  will  be  |<r.  Thus,  when  the  electricity  flows 
from  b  to  c  (Fig.  183),  we  may  suppose  that  it  flows  along  bqt 
and  comes  back  along  qc.  In  the  first  part  of  this  process  heat  is 


422 


A   MANUAL    OF   PHYSICS. 


THERMO-ELECTRICITY.  423 

absorbed ;  in  the  second  part  heat  is  evolved.  In  the  first  part  the 
average  specific  heat  is  (say)  ^M',  where  ^  is  a  constant.  Similarly, 
in  the  second  part,  the  average  specific  heat  is  ^kit,  which  must  be 
regarded  as  a  negative  quantity,  since  heat  is  being  evolved  as  the 
electricity  passes  from  cold  parts  to  hot  parts.  The  total  sum  for 
the  two  metals  is  therefore  ^(\  -  k2]  (V  -  t)=<r1  -  <ra,  so  that  the  whole 
product  (&!-&2)  (t'-f)  (t'-t)  is  identical  with  (<ri-«r2)  (*'-*)•  and 
&!  and  &2  are  double  of  &i  and  &2  respectively. 

The  specific  heat  of  electricity  in  the  metal  whose  line  is  jTi 
(Fig.  183)  is  therefore  represented  by  the  line  qr  at  the  temperature 
t't  and  so  on.  It  is  negative  or  positive  according  as  the  line  slopes 
downwards  (like  that  of  iron,  Fig.  184)  or  upwards  (like  that  of 
copper). 

Le  Eoux  found  that  the  specific  heat  of  electricity  in  lead  is  zero 
(or  very  nearly  so),  and  therefore  lead  is  chosen  as  the  standard  metal 
in  the  construction  of  the  diagram. 

Tait  has  found  the  very  curious  result  that  the  specific  heat  of 
electricity  in  paramagnetic  metals,  such  as  iron  and  nickel,  changes 
sign  at  least  twice  as  the  temperature  is  raised  (see  §  356). 


CHAPTER  XXIX. 

ELECTRIC     CURRENTS. 

334.  Convection  Current  between  Charged  Conductors. — A  pith- 
ball  placed  between  two  oppositely  charged  insulated  conductors, 
and  free  to  move  between  them,  will  gradually  destroy  their  electri- 
fication. When  it  comes  in  contact  with  the  positively  charged 
body,  it  receives  a  positive  charge  with  which  it  moves,  under  the 
action  of  the  electric  force,  towards  the  negatively  charged  body. 
It  gives  its  positive  charge  to  this  body,  and  receives  from  it  a 
negative  charge,  with  which  it  again  moves  towards  the  positive 
conductor,  and  so  on  alternately. 

If  the  charges  of  the  two  conductors  were  originally  equal,  the 
process  will  result  in  the  complete  destruction  of  their  electrification. 
If  only  one  of  the  two  is  originally  charged,  the  process  will  result 
in  the  division  of  the  charge  between  the  two  in  the  ratio  of  their 
capacities. 

Positive  electricity  is  carried  in  one  direction,  and  negative  elec- 
tricity is  carried  in  the  reverse  direction,  by  a  convective  process. 
The  greater  the  electric  force  between  the  conductors  is,  the  more 
rapid  will  be  the  motion  of  the  ball,  and  the  more  nearly  will  the 
process  approach  to  a  continuous  flow  of  electricity. 

335.  Flow  of  Electricity  in  Metallic  Conductors. — If  we  place 
the  two  oppositely  charged  conductors  in  contact  with  each  other  by 
means  of  a  metallic  wire,  a  current  of  electricity  will  be  established 
between  them  until  their  potentials  are  equalised.  If  one  of  the 
bodies  be  electrified,  say  positively,  while  the  other  is  unelectrified, 
the  result  is  (as  above)  that  the  charge  is  divided  between  the  two 
bodies  in  the  ratio  of  their  capacities ;  and  it  is  usual  to  say  that 
positive  electricity  has  flowed  from  the  first  body  to  the  second, 
though  the  same  result  might  have  been  produced  by  a  flow  of  in- 
duced negative  electricity  in  the  opposite  direction. 

Any  difference  of  potential  between  two  parts  of  a  conductor 
constitutes  an  electromotive  force  under  which  transference  of 


ELECTRIC    CURRENTS.  425 

positive  electricity  will  take  place  from  the  part  at  higher  potential 
to  the  part  at  lower  potential.  And  so  long  as  the  potentials  are 
maintained  constant,  the  quantity  of  electricity  which  flows  from 
the  one  part  to  the  other  in  a  fixed  time  remains  constant.  The 
amount  which  flows  per  unit  of  time  along  the  conductor  from  the 
one  part  to  the  other,  is  called  the  Strength  or  Intensity  of  the 
current. 

The  slightest  difference  of  potential  between  two  parts  of  a  con- 
ductor will  produce  a  current  of  electricity  between  these  parts,  but 
no  current  can  be  maintained  without  the  expenditure  of  energy. 
That  is  to  say,  the  flow  of  the  current  is  opposed  by  a  Resistance  in 
the  conductor.  This  is  analogous  to  the  flow  of  a  liquid  along  a 
tube.  A  current  is  produced  in  the  tube  by  the  action  of  the 
slightest  force,  but  work  must  be  suspended  in  order  to  maintain 
the  flow  ;  for  the  motion  is  opposed  by  a  resistance  which  is  due  to 
internal  friction. 

When  an  incompressible  fluid  flows  through  a  tube  because  of  a 
constant  difference  of  pressure  at  its  extremities,  equal  quantities  of 
fluid  pass  every  section  of  the  tube  in  the  same  time.  So  in  the 
flow  of  electricity  along  a  conductor  because  of  a  constant  difference 
of  potential  at  its  extremities  equal  quantities  of  electricity  pass 
every  section  of  the  conductor  in  the  same  time. 

When  by  any  means  a  difference  of  potential  is  maintained  at 
different  parts  of  a  conductor,  we  may  draw  in  the  conductor  a 
series  of  equipotential  surfaces.  The  electric  stream  lines  are 
everywhere  perpendicular  to  these  surfaces. 

Fig.  185  (a)  represents  the  distribution  of  stream  lines  and  equi- 
potential lines  in  a  thin  circular  conducting  sheet,  the  centre  of 
which  is  kept  at  a  constant  negative  potential,  while  a  point  on 
its  circumference  is  kept  at  an  equal  positive  potential.  The  equi- 
potential lines  are  in  part  open  curves  with  their  extremities  on 
the  circumference  of  the  circular  sheet,  and  in  part  they  are  closed 
curves  surrounding  the  centre.  If  the  whole  diagram  be  revolved 
about  the  axis  of  symmetry,  the  various  lines  will  trace  out  surfaces 
of  flow  and  of  constant  potential  in  a  conducting  sphere,  when  the 
given  points  are  maintained  at  constant  (different)  potentials. 

In  the  upper  half  of  Fig.  185  (6)  the  method  by  which  the  stream 
lines  in  the  above  case  are  drawn  is  exhibited.  The  circumference 
of  the  circle  is  divided  into  a  whole  number  of  equal  parts  and  lines 
are  drawn  from  A  and  C  to  the  extremities  of  these  parts.  These 
lines  are  numbered  from  A  round  the  circumference  to  the  opposite 
end  of  the  diameter  AC. 

The  lower  half  of  the  figure  exhibits  the  method  of  drawing  the 


426 


A   MANUAL   OP   PHYSICS. 


equipotential  lines  when  A  and  C  are  points  in  an  infinitely  ex- 
tended conducting  sheet.  The  nature  of  the  difference  between 
them  and  the  equipotential  lines  of  Fig.  185  (a)  is  apparent. 


836.  Ohm's  Law.  Kirclioff's  Laivs. — Ohm  determined  experi- 
mentally the  relation  which  connects  electromotive  force,  current  - 
strength,  and  resistance  in  a  conducting  circuit.  This  relation  is 
therefore  known  as  Ohm's  Law. 

He  found  that  in  a  conductor  of  given  resistance  the  strength  of 
the  current  is  proportional  to  the  electromotive  force,  and  that  in  a 
conductor  of  variable  resistance  the  strength  of  the  current  is 
inversely  proportional  to  the  magnitude  of  the  resistance,  if  the 
electromotive  force  be  constant.  If  E,  C,  and  E  represent  respec- 
tively the  electromotive  force,  the  strength  of  the  current,  and  the 
resistance,  these  results  are  expressed  by  the  equation 

E=CE 

if  we  define  unit  current  as  the  current  which  is  maintained  in  a 
conductor  of  unit  resistance  by  unit  electromotive  force. 

This  law  enables  us  to  calculate  the  resistance  of  a  compound 
conductor  composed  of  a  number  of  conductors  arranged  in  '  series,' 
that  is,  arranged  so  that  the  current  flows  from  one  to  another  of 
the  several  conductors  in  succession.  Let  E  denote  the  total  differ- 
ence of  potential  in  the  circuit,  and  let  E  be  the  total  resistance. 
Also,  let  Ci,  £2,  etc.,  and  rj,r2,  etc.,  ropresent  the  corresponding 


ELECTRIC   CURRENTS.  427 

quantities  for  the  several  conductors  in  the  circuit.    The  same  current 
C  flows  through  all  the  conductors,  and  hence  the  condition 


gives,  by  Ohm's  Law, 


Therefore  the  resistance  of  a  number  of  conductors  arranged  in 
series  is  the  sum  of  their  separate  resistances. 


If  the  conductors  be  arranged  in  '  multiple  arc,'  as  in  Fig.  186, 
the  condition 


where  Cj,  C..,,  etc.,  are  the  currents  in  the  separate  parts  of  the  circuit, 
gives,  by  Ohm's  Law, 

E_E     E 
B    V  V 

whence  —  =  -  -|  ---  j-  .  .  .  . 

B    ra     r2 

The  reciprocal  of  the  resistance  of  a  conductor  is  called  its  con- 
ductivity^ and  so  this  equation  expresses  the  fact  that  the  con- 
ductivity of  a  compound  conductor,  formed  of  a  number  of 
separate  conductors  joined  in  the  multiple  arc  arrangement,  is  the 
sum  of  the  individual  conductivities  of  these  conductors. 

The  law  alluded  to  above,  that  the  flow  of  electricity  in  con- 
ductors resembles  that  of  an  incompressible  fluid,  together  with 
Ohm's  Law,  enables  us  to  express  the  relations  connecting  electro- 
motive, current-strength,  and  resistance  in  the  various  parts  of  any 
network  of  conductors,  however  complex.  These  laws,  stated  in  the 
forms, 

(1)  The  sum  of  the  currents  which  flow  towards  any  point  of 

the  network  is  equal  to  the  sum  of  the  currents  which 

flow  from  it, 


428  A  MANUAL   OF   PHYSICS. 

(2)  The  sum  of  the  electromotive  forces  which  act  in  any  closed 
loop  of  the  network  is  equal  to  the  sum  of  the  products  of 
the  current  into  the  resistance  in  the  several  parts  of  the 
loop, 
are  known  as  Kirchoffs  Laws. 

337.  Electrolytic  Conduction.     Faraday's  Laws. — When  elec- 
tricity passes  through  certain  conductors,  chiefly  liquids,   decom- 
position of  the   conductor   takes   place ;    and   it  appears   that   the 
decomposition  is  a  necessary  accompaniment  of  the  passage  of  the 
electricity.     Such  substances  are  called  electrolytes,  and  the  process 
of  decomposition  is  termed  electrolysis. 

The  current  usually  enters  and  leaves  the  electrolyte  by  metallic 
conductors,  which  are  termed  the  electrodes,  the  conductor  by 
which  the  current  enters  being  distinguished  as  the  anode,  while 
the  conductor  by  which  it  leaves  is  called  the  cathode. 

The  products  of  decomposition  appear  at  the  electrodes,  the 
metallic  constituent  appearing  at  the  cathode,  while  the  other  con- 
stituent appears  at  the  anode.  Thus,  in  the  electrolytic  decomposi- 
tion of  hydrochloric  acid,  hydrogen  appears  at  the  cathode,  while 
chlorine  is  evolved  at  the  anode.  No  decomposition  occurs  in  the 
interior  of  the  electrolyte.  Hence  it  follows  that  when  an  electro- 
motive force  acts  upon  the  electrolyte,  the  metallic  constituent 
travels  in  the  direction  in  which  the  current  goes  (that  is,  from 
high  to  low  potential),  and  the  other  constituent  travels  in  the 
opposite  direction.  The  two  constituents  are  therefore  termed  ions, 
the  one  which  moves  towards  the  cathode  being  called  the  cation, 
while  the  one  which  moves  towards  the  anode  is  called  the  anion. 

The  laws  of  electrolytic  conduction  were  fully  investigated  by 
Faraday.  He  found  that  the  amount  of  the  electrolyte  which  is 
decomposed  by  the  passage  of  a  certain  quantity  of  electricity  is 
strictly  proportional  to  that  quantity.  The  amount  which  is  de- 
composed during  the  passage  of  a  unit  of  electricity  is  called  the 
Electrochemical  Equivalent  of  the  substance. 

Faraday  also  found  that  the  amount  of  any  substance,  which 
appears  as  anion  or  cation,  is  totally  independent  of  the  substance 
with  which  it  is  combined ;  and  this  shows  that  the  electrochemical 
equivalent  of  a  substance  is  absolutely  constant. 

The  process  of  electrolysis  has  received  extensive  practical  appli- 
cations in  the  art  of  electrometallurgy. 

338.  Polarisation.     Ohm's  Law  in  Electrolytes. — The  passage 
of  electricity  through  a  liquid  results  in  the  chemical  decomposition 
of  that  liquid,  and  the  electrical  energy  is  transformed  in  part  into 
potential  energy  of  chemical  separation. 


ELECTRIC    CURRENTS.  429 

Now,  if  two  parts  of  a  conductor  are  kept  at  constant  (different) 
potentials,  the  work  which  is  done  in  transferring  unit  quantity  of 
electricity  from  the  part  at  the  higher  potential  to  the  part  at  the 
lower  potential  is  equal  to  the  difference  of  potential  between  the 
two  parts  —  that  is,  to  the  electromotive  force  (E,  say).  Let  H  be 
the  amount  of  heat  which  is  developed  in  the  combination  of  unit 
amount  of  the  ions,  from  the  state  in  which  they  are  liberated,  so  as 
to  form  the  compound  electrolyte.  This  amount  of  heat  is  equivalent 
to  the  amount  of  work  JH,  where  J  is  the  dynamical  equivalent  of 
heat.  Hence,  in  dynamical  units,  J#H  is  the  work  which  is  ex- 
pended when  unit  amount  of  electricity  passes  through  an  electrolyte 
of  which  q  is  the  electrochemical  equivalent,  and,  therefore, 


The  quantities,  J,  q,  and  H,  in  this  equation  are  all  finite,  and  so 
E  is  finite  ;  and,  therefore,  a  finite  electromotive  force  is  required, 
in  order  to  effect  electrolytic  decomposition. 

If  we  keep  the  electrodes  at  an  insufficient  difference  of  potential, 
no  decomposition  can  ensue  ;  but  if  there  be  any  difference  of 
potential  in  the  circuit,  flow  of  electricity  must  occur.  And  we  are 
here  met  with  a  difficulty,  for  Faraday's  Law  asserts  that  decom- 
position takes  place  in  strict  proportion  to  the  amount  of  electricity 
which  passes.  But  this  difficulty  can  readily  be  explained.  The 
phenomenon  is  precisely  analogous  to  the  charging  of  a  Ley  den  jar. 
k  When  the  jar  is  charged  under  given  conditions  regarding  potential, 
positive  electricity  flows  into  the  one  coating  and  negative  electricity 
flows  into  the  other  until  the  potential  of  the  jar  is  equal  to  that  of 
the  machine  which  is  used  to  charge  it.  But  the  electrical  energy 
is  stored  up  in  the  dielectric  which  insulates  the  coatings  of  the  jar, 
and  will  produce  an  equal  reverse  current  of  electricity  whenever 
the  coatings  are  put  in  metallic  communication  with  each  other, 
and  we  say  that  this  reverse  current  is  produced  by  the  reverse 
electromotive  force  of  the  jar.  Similarly,  when  an  electromotive 
force,  too  feeble  to  cause  decomposition,  acts  upon  an  electrolyte, 
flow  of  electricity  takes  place  in  the  conducting  parts  of  the  circuit 
—  that  is,  in  the  metallic  and  the  electrolytic  conductors.  But  no 
transference  of  electricity  can  take  place  between  the  electrode  and 
the  electrolyte,  for  decomposition  must  occur  (by  Faraday's  law)  if 
it  did,  and  the  electromotive  force  is  not  large  enough  to  produce 
decomposition.  The  layer  (of  thickness  comparable  with  molecular 
dimensions)  between  the  molecules  of  the  electrolyte  and  the  mole- 
cules of  the  electrode  acts  as  an  insulator,  and  its  two  surfaces 
become  oppositely  charged,  as  those  of  a  Leyden  jar  would,  until 


480  A   MANUAL   OF   PHYSICS. 

the  reverse  electromotive  force  produced  in  this  way  becomes 
equal  to  the  direct  electromotive  force.  If  now  the  direct  force  be 
removed  and  the  electrodes  be  joined  so  as  to  form  a  closed  circuit, 
the  reverse  force  will  produce  a  reverse  current. 

This  phenomenon  is  called  polarisation,  the  reverse  force  is 
called  the  electromotive  force  of  polarisation,  and  the  reverse  cur- 
rent is  called  the  polarisation  current. 

Let  E0  be  the  force  which  is  required  to  produce  decomposition. 
If  E  be  the  direct  force,  the  difference  E  -  E0  is  alone  effective  in 
producing  a  permanent  current. 

The  quantity  E0  is  not  constant  except  under  fixed  conditions ; 
and,  in  practice,  the  conditions  under  which  the  decomposition 
takes  place  vary  greatly.  The  electrodes  or  the  electrolyte  may 
have  some  chemical  or  molecular  attraction  for  the  products  of 
decomposition,  and  the  electromotive  force  which  is  required  to 
produce  the  decomposition  will  be  less  in  proportion  as  this  attrac- 
tion is  greater.  As  an  extreme  case,  an  infinitely  small  electro- 
motive force  would  effect  the  decomposition  of  an  electrolyte,  the 
constituents  of  which  had  no  greater  attraction  for  each  other  than 
they  had  for  the  substance  of  the  electrode.  Again,  if  the  products 
of  electrolysis  are  gaseous,  and  are  dissolved  by  the  electrolyte,  de- 
composition will  take  place  more  readily  than  it  would  if  the  gases 
were  not  dissolved.  When  the  process  is  first  started  the  gases  may 
dissolve  readily,  but  their  solution  will  occur  with  greater  and  greater 
difficulty  until  saturation  is  reached,  when  it  attains  a  maximum. 
Further,  if  the  gases  evaporate  from  the  electrolyte,  saturation  will 
never  be  fully  attained,  and  the  maximum  attainable  state'of  satura- 
tion will  depend  upon  the  partial  pressure  of  the  gases  in  the  atmo- 
sphere which  is  in  contact  with  the  electrolyte. 

These  and  other  causes  which  affect  the  electromotive  force  have 
been  fully  discussed  by  Von  Helmholtz,  who  has  given  a  complete 
thermodynamical  theory  of  polarisation,  the  results  of  which  accord 
very  well  with  experimental  facts. 

If  n  be  the  number  of  atoms  in  the  electrochemical  equivalent  of 
a  substance,  the  quantity  qjn  may  be  regarded  as  the  'atomic 
charge,'  which  is  constant  since  q  and  n  are  constant.  This  con- 
stancy of  the  atomic  charge  points  to  an  intimate  relation  between 
electricity  and  matter,  and  the  fact  that  the  electrochemical  equiva- 
lent of  a  substance  is  constant  whether  it  be  electrolysed  from 
combination  with  a  monad,  dyad,  or  triad,  etc.,  element  shows  that 
the  atomic  charge  of  a  dyad  element  is  double  that  of  a  monad 
element,  that  the  atomic  charge  of  a  triad  element  is  three  times 
that  of  a  monad  element,  and  so  on.  This  led  Von  Helmholtz  to 


ELECTRIC    CURRENTS.  431 

regard  chemical  affinity  as  the  result  of  a  greater  attraction  of  some 
atoms  than  of  others  for  positive  or  negative  electricity.  Thus,  for 
example,  oxygen  has  a  greater  attraction  for  negative  electricity 
than  hydrogen  has,  while  hydrogen  attracts  positive  eledhricity  more 
strongly  than  does  oxygen.  And  Von  Helmholtz  regard's  the  affinity 
of  oxygen  and  hydrogen  for  each  other  as  the  result  &t  the  electrical 
attraction  between  positively  charged  hydrogen  and  negatively 
charged  oxygen. 

The  atomic  charge  is  excessively  small  since  n  is  very  large  ;  but, 
since  the  atoms  are  at  an  exceedingly  small  distance  apart,  the 
attraction  may  be  very  great. . 

These  considerations  enabled  Helmholtz  to  explain  a  phenomenon 
which  appeared  to  be  at  variance  with  Faraday's  law.  An  extremely 
feeble  constant  current,  inappreciable  to  all  but  the  most  delicate 
instruments,  flows  in  an  electrolytic  circuit  under  electromotive 
forces  far  too  feeble  to  produce  decomposition.  Von  Helmholtz 
showed  that  this  was  due  to  the  presence  of  dissolved  gases.  Thus 
oxygen  in  solution  will  gradually  find  its  way  to  the  negative 
electrode  and  will  receive  a  negative  charge.  It  then  becomes 
subject  to  the  electromotive  forces  and  travels  to  the  positive  elec- 
trode, to  which  it  gives  up  its  negative  charge.  In  this  way  a 
constant  '  convective '  current  of  electricity  is  kept  up,  and  Faraday's 
law  is  not  subject  to  exception. 

Helmholtz  observed  the  existence  of  this  convective  current 
even  when  he  had?  carefully  freed  the  electrolyte  from  dissolved 
gases  by  the  most  perfect  processes.  Here  again  there  is  no  real 
exception  to  Faraday's  law,  for  Helmholtz's  thermodynamical 
theory  shows  that  some  dissolved  gases  must  be  present  if  the 
(liquid)  electrolyte  is  to  be  in  stable  equilibrium ;  and,  therefore, 
even  if.  all  dissolved  gases  were  abstracted,  decomposition  of  the 
liquid  would  go  on  until  the  requisite  minimum  of  dissolved  gases 
products  were  present. 

In  the  case  of  electrolytes,  Ohm's  law  must  be  put  into  the  form 
E  -E0  =  CK,  E0  being,  as  above,  the  electromotive  force  of  polarisa- 
tion. Its  applicability  has  been  verified,  in  all  cases,  within  the 
limits  of  experimental  error.  The  variability  of  E0  renders  the 
proof  of  the  law  extremely  difficult. 

339.  Production  of  Electric  Currents. — An  electrolytic  circuit 
constitutes  a  system  which  is  in  stable  equilibrium,  for  a  finite 
electromotive  force  is  required  in  order  that  decomposition  may  take 
place,  and  the  removal  of  the  electromotive  force  causes  the  cessation 
of  the  decomposition.  Consequently,  we  conclude  (§  15)  that,  in 
this  stable  system  in  which  the  passage  of  a  divect  current  produces 


432  A   MANUAL    OF    PHYSICS. 

chemical  separation,  the  energy  of  chemical  separation  will  be 
transformed  into  the  energy  of  a  reverse  electric  current,  when  the 
conditions  are  such  that  this  reverse  current  can  flow. 

Therefore,  in  a  closed  conducting  circuit,  which  includes  an 
electrolyte,  and  which  has  associated  with  it  energy  of  chemical 
separation,  the  sum  of  the  electromotive  forces  may  not  be  zero,  but 
will  produce  an  electric  current,  while  chemical  combination  pro- 
ceeds. The  sum  of  the  forces  can  differ  from  zero  only  because 
there  is  a  chemical  source  for  the  energy  of  the  current  which  the 
resultant  force  produces;  just  as,  in  a  thermo-electric  circuit,  the 
electric  energy  is  derived  from  the  heat  energy  which  is  sup- 
plied. 

An  arrangement  of  this  kind  may  therefore  be  used  for  the  pro- 
duction of  an  electric  current ;  and  it  is  of  no  consequence  whether 
the  chemical  separation  is  produced  by  the  previous  passage  of  an 
electric  current  through  the  circuit  under  the  action  of  an  external 
electromotive  force,  or  is  given  as  an  initial  condition  produced  by 
chemical  processes  or  otherwise. 

An  arrangement  of  the  latter  class  constitutes  a  Primary  Cell ; 
one  of  the  former  class  constitutes  a  Secondary  Cell. 

The  equation  in  §  338  indicates  a  method  of  determining  the 
electromotive  force  in  any  such  circuit ;  for  it  asserts  that  '  the 
electromotive  force  of  an  electrochemical  apparatus  is  in  absolute 
measure  equal  to  the  mechanical  equivalent  of  the  chemical  action 
on  one  electrochemical  equivalent  of  the  substance.'  This  was  first 
pointed  out  by  Sir  W.  Thomson.  Conversely,  the  energy  which  is 
developed  in  a  given  chemical  action  can  be  estimated  by  means  of 
measurements  of  the  electromotive  force  of  an  electrochemical 
apparatus  in  which  that  action  is  developed. 

340.  Primary  Cells. — Primary  cells  are  sub-divided  into  two 
classes  according  as  one  fluid  or  two  fluids  are  used  in  their  construc- 
tion. 

All  the  older  cells  were  of  the  one-fluid  type,  and  the  conducting 
circuit  was  generally  made  up  of  a  plate  of  zinc,  a  plate  of  copper, 
and  sulphuric  acid.  The  sulphuric  acid,  combining  with  the  zinc, 
produces  zinc  sulphate,  and  hydrogen  is  liberated  at  the  copper 
electrode.  Positive  electricity  flows  from  the  zinc  to  the  copper 
through  the  liquid  (Fig.  187). 

Such  a  cell  has  many  disadvantages.  Slight  impurities  in  the 
zinc  (possibly  even  differences  in  its  physical  constitution)  at 
different  parts  give  rise  to  local  currents  of  electricity  between  these 
parts,  and  thus  lead  to  solution  of  the  zinc  without  the  production 
of  effective  currents  in  the  main  circuit.  This  local  action  may  be 


ELECTRIC    CURRENTS. 


433 


largely  prevented  by  amalgamating  the  surface  of  the  zinc,  for  the 
surface  is  thus  rendered  more  nearly  homogeneous.  Again,  the 
evolution  of  hydrogen  at  the  copper  electrode  gives  rise  to  a  reverse 


FIG.  187. 

electromotive  force  of  polarisation.  And,  ultimately,  when  the 
sulphuric  acid  is  transformed  into  zinc  sulphate,  zinc,  instead  of 
hydrogen,  is  deposited  upon  the  copper  plate.  When  this  process  is 
complete,  both  electrodes  are  practically  composed  of  zinc  ;  and, 
from  the  symmetry  of  the  arrangement,  it  is  evident  that  the  sum  of 
the  electromotive  forces  in  the  circuit  must  be  zero,  so  that  the 
current  ceases. 

Daniell  improved  this  cell  by  the  introduction  of  two  fluids, 
separated  by  a  porous  diaphragm  which  did  not  prevent  the  passage 
of  electricity.  A  zinc  rod  Z  (Fig.  188)  is  placed  inside  a  porous  cell, 


FIG.  188. 

which  is  contained  in  an  outer  copper  vessel  C.  The  porous  cell 
contains  a  solution  of  zinc  sulphate,  and  the  copper  vessel  contains 
a  saturated  solution  of  copper  sulphate.  In  order  to  keep  this 

28 


434  A   MANUAL    OF    PHYSICS. 

solution  saturated,  crystals  of  the  sulphate  of  copper  may  be  placed 
on  a  perforated  tray  T,  inside  the  copper  vessel.  These  crystals  are 
dissolved  away  when  the  solution  becomes  weakened. 

When  the  cell  is  in  action,  the  current  passes  from  zinc  to  copper 
through  the  liquid.  The  zinc  sulphate  is  electrolysed  into  zinc  and 
'sulphion'  (S04).  The  S04  acts  upon  the  zinc  and  forms  zinc 
sulphate.  Thus  the  zinc  sulphate  always  tends  to  be  saturated. 
The  copper  sulphate  is  simultaneously  electrolysed  into  copper  and 
sulphion,  and  the  copjMr  (being  the  metallic  ion,  and  therefore 
travelling  with  the  current,  which  is  assumed  to  travel  in  the 
direction  in  which  positive  electricity  goes)  is  deposited  upon  the 
copper  vessel.  The  zinc,  which  results  from  the  electrolysis  of  the 
zinc  sulphate,  is  not  deposited  at  all,  but  unites  with  the  sulphion 
produced  by  the  electrolysis  of  the  sulphate  of  copper,  and  so  re- 
forms zinc  sulphate,  which  remains  inside  the  porous  cell.  Thus 
hydrogen  is  not  evolved  at  the  copper  electrode,  and  polarisation  is 
reduced  to  a  minimum. 

If  the  current  be  too  strong,  or  if  the  copper  sulphate  be  not 
saturated,  some  hydrogen  will  be  evolved. 

Daniell's  cell  gives  an  extremely  constant  electromotive  force  if 
the  conditions  which  are  necessary  to  its  proper  action  are  main- 
tained. 

Bunseri's  cell  is  also  a  two-fluid  cell.  A  rod  of  carbon  is  placed 
in  strong  nitric  acid,  which  is  contained  in  the  porous  cell,  and  a 
zinc  plate  is  placed  in  an  outer  cell  of  glazed  earthenware  which 
contains  an  aqueous  solution  of  sulphuric  acid.  Hydrogen  is 
evolved  at  tjie  carbon,  but  is  at  once  oxidised  by  the  nitric  acid. 
The  fumes  which  are  given  off  from  the  acid  are*  very  poisonous,  so 
that  these  cells  should  be  kept  in  a  separate  room,  as  far  as  possible, 
if  a  considerable  number  of  them  are  in  use. 

Grove's  cell  resembles  that  of  Bunsen  in  its  chief  features.  The 
carbon  is  replaced  by  a  sheet  of  platinised  silver.  The  process  of 
platinisation  gives  a  large,  very  finely  corrugated  surface,  which 
favours  the  diminution  of  polarisation. 

The  electromotive  forces  of  the  two  latter  cells  are  very  nearly 
equal  to  each  other,  and  are  considerably  larger  than  that  of  a 
Daniell  cell,  but  are  not  so  constant. 

The  single-fluid  bichromate  cell  (bichromate  of  potassium  dis- 
solved in  an  aqueous  solution  of  sulphuric  acid,  into  which  dip  zinc 
and  carbon  plates)  gives  a  high  electromotive  force,  and  is  greatly 
Used  when  a  powerful  current  is  required  for  short  periods. 

The  Leclanche  cell  consists  of  a  porous  cell  filled  with  manganese 
dioxide,  in  which  a  carbon  rod  is  placed.  A  zinc  rod  is  placed  in  a 


ELECTRIC   CURRENTS. 


435 


solution  of  ammonium  chloride,  outside  the  porous  cell.  The 
electromotive  force  of  this  cell  rapidly  diminishes  when  it  is  made 
to  produce  a  current ;  but  it  soon  attains  its  original  value  after  the 
current  is  stopped.  Its  great  advantage  is  that  it  remains  in  good 
working  order  for  months. 

Many  other  forms  of  cells,  including  modifications  of  the  above, 
are  in  practical  use  ;  but  the  present  examples  will  serve  sufficiently 
as  illustrations  of  their  general  principles. 

The  zinc  and  the  carbon,  or  the  zinc  and  the  copper,  etc.,  are 
called  the  elements  of  the  cell ;  carbon  and  copper  being  positive 
elements,  while  zinc  is  a  negative  element  (for  the  current  flows 
from  carbon  or  copper  to  zinc  through  the  external  conductor). 

Various  cells  may  be  joined  together  to  form  a  voltaic  battery. 
When  the  positive  element  of  one  cell  is  joined  to  the  negative 
element  of  the  next,  the  cells  are  said  to  be  coupled  in  series  ;  or  to 
be  coupled  for  electromotive  force,  for,  in  this  case,  the  electromotive 
force  of  the  battery  is  the  sum  of  the  electromotive  forces  of  the 
several  cells.  When  all  the  positive  elements  are  joined  together, 
and  all  the  negative  elements  are  joined  together,  the  cells  are  said 
to  be  coupled  for  quantity,  for  the  whole  battery  acts  as  one  large 
cell  of  the  same  kind,  and  produces  a  powerful  current. 

341.  Secondary  Cells. — Grove's  Gas  Battery  was  the  earliest 
form  of  secondary  cell.  It  consists  essentially  of  two  glass  tubes 
(A,  B,  Fig.  189),  which  are  fitted  into  the  two  necks  of  an  ordinary 
Woulfe's  bottle.  These  tubes  are  open  at  their  lower  ends,  but  are 


FIG.  189. 

closed  at  their  upper  ends.  Platinum  wires  are  inserted  through  the 
glass  at  the  closed  ends,  and  are  welded  to  strips  of  platinum  foil 
which  extend  throughout  nearly  the  whole  length  of  the  tubes. 
The  tubes  and  the  bottle  are  filled  with  (say)  dilute  solution  of 
sulphuric  acid. 

28—2 


486  A    MANUAL    OF    PHYSICS, 

If  a  current  be  sent  round  the  circuit  from  A  to  B  through  the 
liquid,  oxygen  will  be  evolved  in  A  and  hydrogen  will  be  evolved  in 
B  ;  and  a  reverse  electromotive  force  will  be  produced.  If  the 
direct  current  be  stopped,  and  if  the  wires  let  into  A  and  B  ba 
joined,  this  reverse  force  will  produce  a  reverse  current  setting  from 
B  to  A  through  the  liquid.  The  liquid  will  be  decomposed,  oxygen 
tending  to  appear  in  B  and  hydrogen  tending  to  appear  in  A.  The 
gases  are  not  really  set  free ;  they  combine  with  the  gases  already 
in  the  tubes  to  form  water ;  and  this  process  goes  on  until  both 
tubes  are  again  filled  with  the  liquid  solution. 

[It  is  interesting  to  note  that  the  arrangement  may  be  used  as  a 
primary  battery.  Instead  of  evolving  the  gases  by  a  direct  current, 
we  may  introduce  them  independently  into  the  tubes,  and  the 
current  will  proceed  from  B  to  A  through  the  liquid  as  formerly. 

The  current  will  flow  even  if  hydrogen  be  introduced  into  B  while 
A  is  left  full  of  liquid.  The  hydrogen  in  B  unites  with  the  oxygen 
which  is  produced  by  the  electrolytic  action  of  the  current,  and 
hydrogen  is  evolved  in  A.  The  sum  of  the  electromotive  forces  in 
the  circuit  soon  becomes  zero,  since  there  is  hydrogen  in  both  tubes, 
and  the  current  stops. 

If  A  be  filled  with  ordinary  oxygen  while  B  is  left  full  of  liquid,, 
very  little  effect  will  be  observed;  and  this  shows  that  electro- 
lytically  evolved  oxygen  differs  considerably  in  its  properties  from 
ordinary  oxygen.  If  A  contains  ozone,  the  action  will  proceed  as 
before.] 

Planters  secondary  cell  consists  of  two  large  sheets  of  lead, 
separated  by  a  sheet  of  guttapercha,  and  rolled  up  into  a  spiral 
form.  These  sheets  are  placed  in  a  vessel  which  contains  dilute 
sulphuric  acid,  and  an  electric  current  is  passed  through  the  cell 
from  one  sheet  of  lead  to  the  other.  The  oxygen  which  is  evolved 
at  the  one  sheet  forms  a  layer  of  peroxide  of  lead  on  its  surface. 
The  direct  current  being  stopped,  and  the  terminals  of  the  lead  sheets 
being  joined,  a  reverse  current  flows  through  the  cell.  The  hydrogen 
which  is  now  evolved  partly  reduces  the  peroxide  of  lead,  and  the 
oxygen  unites  with  the  lead  of  the  other  electrode  to  form  peroxide ; 
and,  when  both  electrodes  become  similar,  the  current  stops.  A 
current  is  now  passed,  from  an  external  source,  through  the  cell  in 
the  same  direction  as  that  in  which  the  reverse  current  flowed. 
When  the  gases  bubble  off  freely  at  the  electrodes,  this  current  is 
stopped,  and  'the  polarisation  current  is  allowed  to  flow ;  and  so  on 
alternately  for  a  number  of  times.  This  process,  which  is  said  to 
form  the  cell,  gradually  changes  the  lead  into  a  spongy  condition, 
and  thus  greatly  increases  the  charge  which  the  cell  can  contain. 


ELECTRIC    CURRENTS.  487 

When  the  process  of  formation  is  complete,  the  cell  is  always 
charged  by  a  current  in  one  direction  only. 

•  Fame's  cell  is  essentially  similar  to  Planted,  but  the  plates  of 
lead  are  at  first  covered  with  a  coating  of  a  lower  oxide  of  lead. 
"When  the  direct  current  is  passed,  the  oxide  on  the  one  plate  is 
changed  to  the  peroxide,  while  that  on  the  other  is  reduced  to  the 
metallic  condition.  This  avoids  the  tedious  process  of  forming  the 
cell  by  alternate  currents  in  opposite  directions. 

The  modern  accumulator  is  constructed  on  essentially  the  same 
principles  as  the  Faure  cell  is  constructed  on,  though  various  im- 
provements are  introduced.  Its  electromotive  force  exceeds  that  of 
a  Bunsen  cell,  while  its  resistance  is  extremely  small,  so  that  it  is 
capable  of  producing  a  very  powerful  current. 

342.  Transformations  of  Electric  Energy  in  Conducting 
Circuits. — If  the  potential  V  of  a  conductor  be  kept  constant  while 
its  charge  alters  by  the  amount  Q,  the  electric  energy  changes  by 
the  amount  VQ.  Hence,  if  the  quantity  Q  of  electricity  flows  from 
a  part  of  a  conductor  where  the  potential  is  V  to  a  part  where  the 
potential  is  V,  the  energy  expended  in  the  production  of  this  flow  is 
(V-V')Q»  that  is,  it  is  EQ,  where  E  is  the  electromotive  force 
which  acts  between  the  given  parts.  Hence,  if  a  current  of  intensity 
C  is  maintained  between  these  parts,  the  rate  at  which  energy  is  ex- 
pended in  the  process  is  EC. 

This  energy,  which  is  associated  with  the  current,  will  be  trans- 
formed into  heat  in  the  circuit,  if  no  other  transformations  take 
place.  If  a  reverse  electromotive  force  E0  acts  in  the  circuit,  work 
is  expended  at  the  rate  E0C  in  making  the  current  flow  in  a  direc- 
tion opposite  to  that  in  which  the  reverse  force  acts.  And  this 
energy  is  transformed  into  heat  at  the  place  where  the  reverse  force 
acts.  (The  Peltier  evolution  of  heat  at  a  thermo-electric  junction  is 
a  case  in  point.) 

If  no  part  of  the  direct  electromotive  force  acts  against  a  reverse 
force,  and  if  R  is  the  resistance  of  the  circuit,  Ohm's  law  gives 
E  =  CB,  whence  we  see  that  the  rate  at  which  heat  is  developed  in 
the  circuit  is,  in  electrical  units, 

EC2. 

that  is  to  say,  the  rate  at  which  heat  is  developed  in  a  circuit  is 
directly  proportional  to  the  resistance  of  the  circuit  and  to  the 
square  of  the  intensity  of  the  current  which  flows  through  it. 
This  is  known  as  Joule's  Law. 

Part  of  this  heat  is  produced  in  the  cell  which  is  used  for  the  pro- 
duction of  the  current,  but  by  making  the  external  resistance  very 
large  in  comparison  with  the  internal  resistance  of  the  cell,  this 


438  A   MANUAL    OF    PHYSICS. 

portion  may  be  made  as  small  a  fraction  of  the  total  amount  as  we 
please. 

This  principle  is  utilised  in  the  process  of  electric  lighting.  The 
carbon  filament  of  an  '  incandescent '  lamp  is  of  relatively  large  re- 
sistance. Almost  all  the  heat  into  which  the  electric  energy  is 
transformed  is  developed  in  the  carbon,  which  becomes  '  white-hot.' 
In  the  '  arc '  lamp  the  chief  part  of  the  resistance  is  in  the  air-gap 
between  the  carbon  poles,  and  the  heat  which  is  developed  raises  the 
air  to  so  high  a  temperature  that  it  becomes  intensely  luminous. 
[Air  at  ordinary  temperatures  is  an  insulator,  but  hot  air  admits  of 
the  passage  of  electricity  through  it  with  comparative  ease.  The 
heating  of  the  air  is  effected  by  allowing  the  carbons  to  touch  each 
other,  so  that  the  current  flows  through  them  and  makes  them  red- 
hot  near  the  point  of  contact.  When  this  results,  the  carbons  may 
be  separated  to  a  slight  extent,  and  the  current  will  continue  to  flow.] 

The  energy  of  an  electric  current  may  be  directly  transformed 
into  mechanical  work  by  means  of  an  electro-motor  (§  366). 

343.  Measurement  of  Electromotive  Force,  Current  Strength 
and  Resistance. — The  electromotive  force  which  acts  between  two 
parts  of  a  conductor  may  be  determined  directly  by  means  of  the 
electrometer  (§  325),  for  the  problem  is  merety  one  of  the  determina- 
tion of  difference  of  potential. 

The  strength  of  the  current  which  flows  in  a  circuit  may  be  found 
directly  by  placing  an  electrolytic  cell  (or  voltameter)  in  the  circuit. 
The  quantuVy  of  electricity  which  passes  through  the  cell  per  unit  of 
time  is,  by  Faraday's  law,  directly  proportional  to  the  amount  of 
chemical  decomposition  which  takes  place  per  unit  of  time.  [The 
arrangement  which  was  described  as  Grove's  gas  battery,  in  §  341 
constitutes  a  voltameter,  and  the  amount  of  oxygen  or  of  hydrogen 
which  is  evolved  per  unit  of  time  in  either  inverted  tube  can  be 
measured  directly.  If  Q  be  this  quantity,  while  q  is  the  electro- 
chemical equivalent  of  the  substance,  the  strength  of  the  current  is  Q/#.] 


FIG.  190. 

In  practice  the  galvanometer  is  more  frequently  used  for  the 
measurement  of  the  strength  of  a  current.    It  consists  essentially 


ELECTRIC    CURRENTS.  439 

(Fig.  190)  of  a  coil  of  wire  within  which  a  magnet  is  freely  sus- 
pended. The  magnet  lies  normally  in  the  pmne  of  the  coil,  but 
when  a  current  flows  round  the  coil,  the  magnet  tends  to  place  its 
length  at  right  angles  to  the  plane  of  the  coil.  This  tendency  is 
opposed  by  the  action  of -a  constant  external  magnetic  force,  and  the 
tangent  of  the  angle  of  deflection  of  the  magnet  is  proportional 
to  the  strength  of  the  current  (§  369). 

The  resistance  of  a  wire  may  be  determined  in  terms  of  any  given 
unit  by  means  of  the  arrangement  known  as  Wheatstone's  Bridge. 
Let  four  conductors,  the  resistances  of  which  are  rlt  ra,  r3,  and  r4,  be 
arranged  as  in  Fig.  191,  and  let  a  battery  b  be  placed  between  the 
points  A  and  B,  so  that  currents  flow  (say)  in  the  way  which  is 
indicated  by  the  arrows.  The  potential  at  the  point  C  is  inter- 


mediate between  those  of  A  and  B,  since  the  current  flows,  through 
C,  from  A  to  B.  And,  by  Ohm's  law,  since  the  same  current  flows 
along  AC  and  CB,  the  potential  at  C  must  divide  the  difference  of 
potential  between  A  and  B  in  the  ratio  of  TI  to  r2.  Similarly  the 
potential  at  D  will  divide  that  difference  in  the  ratio  of  r3  to  r4. 
Hence,  if  C  and  D  are  at  the  same  potential  (which  may  be  deter- 
mined by  the  fact  that  no  current  will  then  flow  through  a  galvano- 
meter which  is  placed  between  C  and  D),  the  resistances  must  be 
connected  by  the  relation 


And  consequently,  if  we  know  the  ratio  ra/r4,  and  also  the  absolute 
value  of  r2,  we  can  calculate  the  value  of  r^ 

If  we  know  the  value  of  any  two  of  the  three  quantities — electro- 
motive force,  current -strength,  and  resistance — we  can  calculate  that 
of  the  third  by  means  of  Ohm's  law  (E=CE).  Or,  if  H  be  the  heat 
which  is  developed  in  a  conductor  by  transformation  of  electric 
energy,  while  J  is  the  mechanical  equivalent  of  heat,  we  can  calcu- 


440  A    MANUAL    OF    PHYSICS. 

late  the  values  of  any  two  of  the  three  quantities,  when  we  know 
that  of  the  third,  by  means  of  Joule's  law, 


The  resistance  of  a  metallic  conductor  increases  when  its  tempera- 
ture is  raised,  but  the  resistance  of  an  electrolytic  conductor 
diminishes  when  its  temperature  increases.  The  law  of  variation 
being  known,  we  can  determine  temperature  by  means  of  measure- 
ments of  resistance.  This  method  is  very  useful  when  the  tempera- 
ture is  high. 

A  discussion  of  the  various  units  in  terms  of  which  electrical 
quantities  are  measured  will  be  given  in  Chap.  XXXI. 

The  variation  of  resistance  with  temperature  forms  the  basis  of 
the  most  delicate  method  for  the  measurement  of  radiant  energy. 
The  bolometer  which  is  used  for  this  purpose  consists  essentially  of 
an  extremely  sensitive  and  well-balanced  Wheatstone's  Bridge. 
This  bridge  is  thrown  off  balance,  and  a  current  is  produced  through 
the  galvanometer  when  radiant  heat  falls  on  one  of  the  resistances. 


CHAPTEE  XXX. 

MAGNETISM. 

344.  Fundamental  Phenomena. — Certain  bodies,  when  they  are 
suspended  in  such  a  way  as  to  be  able  to  turn  in  any  direction,  are 
found  to  have  a  marked  tendency  to  place  a  definite  set  of  lines 
drawn  in  their  substance  parallel  to  a  definite  direction  in  space. 
Such  bodies  are  said  to  be  magnetised,  and  are  called  'magnets. 

This  property  of  magnetisation  is  possessed  notably  by  one  of  the 
oxides  of  iron  (lodestone),  but  it  may  be  induced  to  a  much  greater 
extent  in  pieces  of  steel  or  metallic  iron.  The  metals  cobalt  and 
nickel  are  also  capable  of  becoming  strongly  magnetic.  All  other 
substances  have  relatively  extremely  feeble  magnetic  properties. 

Magnets  are  classified  as  permanent  or  temporary ',  according  as 
they  do  or  do  not  retain,  in  large  part  at  least,  their  state  of  mag- 
netisation after  the  removal  of  the  influence  which  caused  it  to  be 
manifested.  A  bar  of  steel  is  of  the  former  kind,  a  bar  of  soft  iron 
is  of  the  latter  kind. 

Any  magnetic  substance,  when  it  is  placed  in  the  immediate 
neighbourhood  of  a  magnetised  body  (more  generally  in  a  field  of 
magnetic  force,  see  §  362),  becomes  magnetised  and  retains  its 
magnetisation  so  long  as  it  remains  in  that  position.  Whether  or 
not  it  will  retain  its  magnetisation  after  removal  from  the  neigh- 
bourhood of  the  magnetised  body  depends  upon  its  physical  consti- 
tution and  upon  circumstances  which  will  be  considered  afterwards. 
And  further,  the  substance  in  which  magnetisation  is  thus  induced 
will  (in  general)  be  attracted  by  the  magnetised  body. 

For  the  sake  of  definiteness,  let  us  consider  the  action  of  an 
ordinary  '  bar '  magnet,  i.e.,  a  permanent  magnet  made  of  a 
rectangular  or  cylindrical  bar  of  steel.  If  all  similar  parts  of  this 
bar  are  similarly  magnetised  (a  condition  which  is  not  realisable  in 
practice),  or  if  it  be  symmetrically  magnetised  with  regard  to  its 
axis  of  figure,  it  will,  when  freely  suspended,  place  its  axis  of  figure 
in  the  definite  direction  in  space  above  alluded  to.  One  definite  end 
of  the  magnet  will  point,  on  the  whole,  northwards,  the  other  will 


442  A    MANUAL    OF    PHYSICS. 

necessarily  point  southwards.  [The  magnet,  if  turned  round 
exactly  end  for  end,  may  remain  in  the  reverse  position  for  a  brief 
time ;  but  it  is  essentially  in  unstable  equilibrium,  and  will,  if  dis- 
turbed to  the  slightest  extent  from  rest,  turn  round  into  its  normal 
position.] 

The  suspended  magnet  will  depart  from  this  normal  attitude  if 
we  bring  up  another  magnet  into  its  neighbourhood.  Those  ends  of 
the  two  which  naturally  point  northwards  appear  to  repel  each  other. 
Those  which  point  southwards  also  exhibit  mutual  repulsion ;  whilst 
those  which  naturally  point  oppositely  appear  to  attract  each  other. 

345.  'North'  and  'South'  Magnetism. — The  phenomena  which 
we  have  just  considered  present  obvious  analogies  to  electrostatic 
phenomena.  Two  electrostatic  systems,  each  consisting  of  two 
oppositely  charged,  insulated,  rigidly-connected,  conducting  spheres, 
placed  near  each  other  in  a  uniform  field  of  electrostatic  stress, 
would  exhibit  similar  mutual  action,  and  would,  when  free  from 
each  other's  influence,  take  up  a  position  in  which  the  line  joining 
the  centres  of  the  insulated  spheres  coincided  in  direction  with  the 
lines  of  electrostatic  force  in  the  surrounding  medium.  Hence,  by 
analogy,  we  may  assume  that  there  are  two  kinds  of  magnetism ; 
that  like  kinds  repel  each  other;  that  unlike  kinds  attract  each 
other  ;  and  that  the  force  of  attraction  or  repulsion  diminishes  as 
the  distance  between  the  attracting  or  repelling  bodies  increases. 

It  is  usual  to  call  the  magnetism,  which  is  found  at  that  end  of 
a  magnet  which  points  northwards,  north  magnetism;  while  the 
opposite  kind  is  called  south  magnetism.  But,  since  it  is  usual  to 
distinguish  the  ends  of  a  magnet  by  colouring  the  north-pointing 
end  red  and  the  south -pointing  end  blue,  the  terms  red  and  blue 
magnetism  are  sometimes  used  instead  of  these,  though  their  use 
cannot  be  commended. 

Further,  just  as  the  action  of  an  electrified  body  separates  the 
neutral  electricities  in  an  adjacent  conductor,  we  might  expect  by 
analogy  that  a  magnetic  substance  would  become  magnetised  so 
long  as  it  remained  in  the  neighbourhood  of  a  magnet ;  and  that  it 
would  be  attracted  towards  the  magnet  just  as  the  conductor  would 
be  attracted  to  the  electrified  body.  All  these  results  happen,  but  it 
is  not  well  to  push  the  analogy  too  far.  Thus,  while  the  conductor 
ceases  to  exhibit  electrification  when  it  is  removed  from  the  influ- 
ence of  the  electrified  body,  a  magnetic  body  will  not  necessarily 
(or  even  generally)  cease  to  exhibit  magnetisation  when  it  is 
removed  from  the  influence  of  the  magnet.  And  it  must  be 
remembered  that  the  phrase  '  two  kinds  of  magnetism '  is  merely 
adopted  as  a  matter  of  convenience.  (Cf.  §  308.) 


MAGNETISM.  443 

346.  Paramagnetic  and  Diamagnetic  Bodies. — Another  point  in 
which  the  direct  analogy  between  electrical  and  magnetic  action 
breaks  down  is  in  the  repulsion  of  some  bodies  from  a  magnet. 
Let  us  suppose  that  the  magnet  is  so  long  that  the  magnetic  body 
which  we  are  considering  is  subject  only  to  the  action  of  the  mag- 
netism at  the  near  end  of  the  magnet.     Under  this  condition  some 
bodies  are  attracted  to  the  magnet,  while  others  are  repelled  from 
it.     Bodies  of  the  former  kind   are  called  paramagnetic  bodies ; 
those  of  the  latter  kind  are  called  diamagnetic  bodies. 

There  is  nothing  in  electricity  corresponding  to  diamagnetism. 

347.  Magnetism  a  Molecular  Phenomenon. — The  great  distinc- 
tion between  electrical  and  magnetic  phenomena  lies  in  the  absence 
of  anything  of  the  nature  of  conduction  of  magnetism.     While  it 
might  seem  that  the  disappearance  of  induced  magnetisation,  when 
a  piece  of  soft  iron  is  withdrawn  from  the  neighbourhood  of  a 
magnet,  is  due  to  the  flowing  together  of  the  two  opposite  kinds 
of  magnetism,  the  persistence  of  magnetisation  to  an  appreciable 
extent  when  the  soft  iron  is  replaced  by  hard  steel  at  once  disposes 
of  this  view. 

And,  further,  if  we  bring  a  magnetic  substance  into  contact 
with  one  end  of  a  magnet  and  then  withdraw  it  from  contact,  no 
interchange  of  magnetism  takes  place,  though  an  interchange  of 
electricity  would  occur  if  the  substances  were  electrified  conductors. 
Also,  while  a  conductor  under  the  influence  of  an  electrified  body 
may  be  divided  into  two  oppositely  charged  portions,  it  is  impossible 
to  divide  a  magnet  into  two  oppositely  magnetised  portions — that  is 
to  say,  it  is  impossible  to  isolate  one  kind  of  magnetism. 

Every  portion,  however  small  it  may  be,  into  which  a  magnet 
may  be  broken  exhibits  properties  precisely  similar  to  those  which 
were  manifested  by  the  complete  magnet.  We  conclude,  therefore, 
that  this  would  still  hold  if  the  magnet  were  reduced  to  its  con- 
stituent molecules ;  that  each  molecule  of  a  magnetised  body  is 
itself  a  little  magnet. 

It  is  easy  to  explain,  upon  this  assumption,  how  it  is  that  mag- 
netisation is  only  evident  near  the  ends  of  a  magnet.  For,  if  the 


FIG.  192. 

little  circles  in  Fig.  192  represent  the  magnetised  molecules,  we  see 
that  the  effect  of  any  north  end  of  a  molecule  at  external  points  is 


444  A    MANUAL    OF    PHYSICS. 

counterbalanced  by  the  effect  of  the  south  end  of  an  immediately 
adjacent  molecule.  It  is  only  at  the  extremities  of  such  a  chain  of 
molecules  that  the  magnetisation  can  become  manifest  through  the 
production  of  external  effects,  and  the  magnetism  is  of  opposite 
kinds  at  the  two  extremities  of  the  chain. 

348.  The  Law  of  Magnetic  Attraction  and  Repulsion.  —  We  can 
investigate,  by  methods  to  be  discussed  subsequently  •(§  358),  the 
law  of  attraction  or  repulsion  between  the  quantities  of  magnetism 
which  we  assume  to  exist  at  the  ends  of  magnets.  The  results  of 
such  measurements  make  it  evident  that  the  force  between  two 
quantities  is  directly  proportional  to  the  magnitude  of  each  quantity, 
and  is  inversely  proportional  to  the  square  of  the  distance  by  which 
they  are  separated.  If  we  choose  to  regard  an  attractive  force  as 
negative,  and  a  repulsive  force  as  positive,  we  can  symbolise  this 
law  by  the  equation 


where  q,  q',  represent  the  quantities  of  magnetism,  and  s  is  the  dis- 
tance between  them  ;  for  F  is  positive  or  negative  according  as  q 
and  q'  are  of  like  or  of  opposite  signs. 

This  law  is  identical  in  form  with  the  law  of  electrical  attraction 
or  repulsion,  and  hence  all  the  results  which  we  have  previously 
deduced  in  the  theory  of  electrostatics  are  capable  of  direct  applica- 
tion in  magnetostatics. 

349.  Poles,  Axis,  and  Magnetic  Moment  of  a  Magnet.—  Those 
two  points  of  a  magnet,  at  which  its  north  and  south  magnetisms 
may  be  supposed  to  be  concentrated,  in  order  to  produce  the  same 
effects  at  external  points  as  the  actual  distribution  of  magnetism 
produces,  are  called  the  Poles  of  the  magnet  ;  and  the  line  joining 
the  poles  is  called  its  Axis. 

In  the  case  of  a  uniformly  magnetised  (rectangular  or  cylindrical) 
bar  magnet,  the  poles  would  be  at  the  geometrical  centre  of  the 
ends  of  the  magnet,  and  the  axis  would  coincide  with  the  axis  of 
figure.  In  any  actual  magnet  the  poles  are  not  exactly  at  the 
ends. 

The  quantity  of  north  magnetism  at  the  one  pole  of  a  magnet,  or 
the  (equal)  quantity  of  south  magnetism  at  the  other  pole,  is  called 
the  Strength  of  the  pole.  The  product  of  the  strength  into  the 
distance  between  the  poles  is  called  the  Magnetic  Moment  of  the 
magnet.  It  is  obviously  analogous  to  the  moment  of  a  couple 
••(§.70). 

350-  Lines  of  Magnetic  Force.     Magnetic  Potential  —  A  region 


MAGNETISM. 


445 


throughout  which  magnetic  force  is  manifested  is  called  a  Field  of 
Magnetic  Force,  or,  more  shortly,  a  magnetic  field.  And  we  may 
imagine  this  field  to  be  filled  with  lines  of  magnetic  force  drawn  in 
the.  direction  in  which  a  north  pole  would  be  moved.  If  we  draw 
from  any  magnetic  pole  a  number  of  lines  of  force  numerically 
equal  to  4;r  times  the  strength  of  the  pole,  the  number  of  these 
which  cross  unit  of  area  of  any  plane  surface  passing  through  any 
point  can  be  made  to  represent  the  strength  of  the  field — i.e.,  the 
magnitude  of  the  force — at  that  point  in  the  direction  of  the  normal 
to  the  given  plane.  In  fact,  as  we  have  already  seen,  all  the  results 
previously  given  regarding  electric  lines  of  force  can  be  at  once 
applied  to  magnetic  lines  of  force,  and  we,  therefore,  do  not  require 
to  repeat  them  here.  It  is  merely  necessary  to  replace  the  term 
'  electrified  body '  by  the  term  '  magnetised  body,'  the  term  '  positive 
electricity '  by  the  term  'north  magnetism,'  'negative  charge'  by 
'  quantity  of  south  magnetism,'  and  so  on. 

A  line  of  force  can  be  readily  traced  out  by  means  of  a  very 
small  magnet,  freely  suspended,  which  is  always  moved  in  the 
direction  in  which  it  points.  In  every  position  its  length  is  tan- 


FIG.  193. 

gential  to  the  line  of  force  which  passes  through  its  centre.  The 
lines  of  force  due  to  any  group  of  magnets  can  also  be  readily  shown 
by  means  of  iron  filings  dusted  over  a  sheet  of  paper,  which  is 
placed  over  the  magnets.  The  filings  become  magnetised  and  turn 
so  as  to  place  their  lengths  in  the  direction  of  the  force.  A  slight 


446  A   MANUAL    OF   PHYSICS. 

tapping  of  the  paper  will  cause  the  filings  to  group  themselves  in 
definite  lines,  each  of  which  coincides  with  a  line  of  force.  The 
vibration  of  the  paper  throws  the  filings  up  into  the  air  for  a 
moment,  so  that  they  are  free  to  accommodate  themselves  to  the 
influence  of  neighbouring  filings.  (See  Figs.  193,  194.) 


FIG.  194. 

Following  the  electrostatic  analogy,  we  may  define  the  Magnetic 
Potential  at  any  point,  due  to  a  magnetic  pole,  as  the  work  which 
is  expended  in  bringing  a  unit  north  pole — that  is,  a  north  pole  of 
unit  strength — from  an  infinite  distance  to  that  point.  The  results 
already  deduced  regarding  electrostratic  potential  will  then  apply 
directly  to  our  present  subject. 

351.  Magnetic  Intensity.  Magnetic  Induction. — If  a  rectangular 
bar-magnet  were  uniformly  magnetised  in  the  direction  of  its  length 
(say),  it  is  obvious  that  its  total  magnetic  moment  is  equal  to  the 
sum  of  the  moments  of  any  number  of  parts  (uniformly  magnetised 
in  the  same  manner),  into  which  we  may  suppose  it  to  be  divided. 
For,  if  L  =  Z1+Z2+  •  •  •  •  +Z-  be  the  total  length  of  the  magnet,  and 
if  Q  be  its  pole -strength,  we  have 

LQ=(Z1  +  /3+ +Z.)Q=?iQ+ZaQ+  ....  +Z.Q, 

which  proves  the  proposition  so  far  as  transverse  division  is  con- 
cerned. And,  since  the  magnetisation  is  uniform,  if  we  divide  it 
longitudinally,  each  part  becomes  a  magnet,  whose  pole-strength  is 


MAGNETISM.  447 


proportional  to  the  area  of  its  end.  Hence,  if  A.  =  a1  +  a2+  .  .  .  .  an 
be  the  total  area  of  the  end,  and  if  3E  be  the  strength  per  unit  of 
area,  so  that  fiA  =  Q  =  E(ai+«2-f  ....  H-^»)  =  ?i+92+  -  •  •  >  +9« 
where  qlt  etc.,  are  the  strengths  of  the  several  parts,  we  have 


which  proves  the  statement  for  longitudinal  division. 

But  it  is  obvious  that  the  statement  is  true  whatever  be  the  forms 
of  the  parts  into  which  the  magnet  is  divided,  for  each  part  may  be 
supposed  to  be  built  up  of  an  infinitely  great  number  of  infinitely 
small  rectangular  portions.  And  it  follows  from  this  consideration 
also  that  the  proposition  is  true  whatever  be  the  form  of  the  original 
magnet. 

The  quantity  I,  the  pole-strength  per  unit  of  area,  is  called  the 
Intensity  of  Magnetisation  of  the  given  magnet.  It  is  evident 
that  we  may  regard  it  as  being  the  magnetic  moment  per  unit  of 
volume. 

Let  us  imagine  a  cylindrical  crevasse  to  be  cut  out  in  the  interior 
of  a  uniformly  magnetised  body.  Let  it  be  bounded  by  plane 
surfaces  perpendicular  to  the  direction  of  magnetisation  ;  and,  while 
all  its  dimensions  are  infinitely  small,  let  the  perpendicular  distance 
between  these  planes  be  infinitely  smaller  than  the  transverse 
dimensions.  The  surface  density  of  magnetism  on  the  plane  faces 
of  the  cavity  is  3E,  north  magnetism  being  distributed  on  the  plane 
face  next  the  south  pole  of  the  magnet,  while  south  magnetism  is 
distributed  on  the  other  ;  and  hence  the  force  in  the  space  between 
the  planes  is  47rH  (§  99).  We  may  therefore  suppose  that  47r$  lines 
of  force  are  drawn  per  unit  of  area  across  this  cavity  in  the  direc- 
tion of  magnetisation  (that  is,  from  the  south  pole  to  the  north  pole 
within  the  substance  of  the  magnet).  It  is  customary  to  call  these 
lines  the  Lines  of  Magnetisation. 

The  lines  of  magnetisation  do  not  constitute  all  the  lines  of 
force  in  the  interior  of  the  magnet.  There  may  be  lines  of  force 
due  to  external  magnetisation.  This  distribution  of  force  must  be 
investigated  in  precisely  the  same  manner  as  that  in  which  we 
investigate  the  distribution  of  force  outside  a  magnet. 

Let  us  suppose  that  the  cylindrical  cavity  is  infinitely  long  in 
comparison  with  its  cross  dimensions.  The  magnetisation  at  the 
ends  of  this  cavity  exerts  no  effect  upon  a  point  at  its  centre,  and 
hence  any  force  found  at  the  centre  must  be  due  to  external  magne- 
tisation. This  quantity  is  denoted  by  the  symbol  jfy.  and  is  called 
the  magnetic  force  at  the  point.  The  total  force,  IB,  is  called  the 


448  A   MANUAL   OF   PHYSItS. 

Magnetic  Induction  at   the   given   point   in   the   magnet,   and   is 
equal  to 


It  is  usual  to  call  the  total  lines  of  force  inside  a  magnet,  Lines  of 
Induction.  They  consist,  therefore,  partly  of  lines  of  magnetisa- 
tion, and  partly  of  the  lines  of  force  within  the  uncut  magnet.  They 
are  continuous  with  the  lines  of  force  external  to  the  magnet. 

[It  must  be  remembered  that  the  three  quantities  13,  fi,  and  1|, 
are  vector  quantities,  and  are  therefore  subject  to  the  laws  of  vector 
addition  (§  40).  In  most  practical  cases,  however,  IE  and  1$  are 
either  similarly  or  oppositely  directed.] 

The  force  due  to  the  magnetism  at  the  surface  of  the  magnetised 
body  is  included  in  the  quantity  H?,  and  is  obviously  directed 
oppositely  to  E  since  it  acts  in  the  direction  of  a  line  drawn  from  the 
north  pole  to  the  south  pole  through  the  material  of  the  body.  It 
therefore  acts  so  as  to  demagnetise  the  body,  and  has  its  greatest 
value  27rl£  at  points  close  to  the  ends  of  the  magnet  (Compare  §  99). 
[To  obviate  demagnetisation,  bar  magnets,  when  not  in  use,  are 
placed  parallel  to  each  other  with  their  like  poles  oppositely  directed, 
and  '  keepers  '  made  of  a  magnetic  metal  (soft  iron  preferably)  are 
placed  in  contact  with  their  ends  (Fig.  195).  A  closed  magnetic 


S  N 

S 

N 
S 

NS 

N 

FIG.  195. 

circuit  is  thus  formed,  for  the  effect  of  the  magnetism  at  the  poles 
is  annulled  by  that  of  the  magnetism  which  is  induced  in  the 
keepers.] 

It  appears,  therefore,  that  the  form  of  a  magnet  must  have  an 
effect  upon  the  distribution  of  magnetisation  throughout  its  interior. 
Thus,  while  in  a  long  rod  placed  parallel  to  the  direction  of  the 
force  in  a  uniform  field,  the  magnetisation  is  sensibly  uniform 
except  near  the  ends,  in  a  short  rod  the  magnetisation  is  very  far 
from  being  uniform. 

The  introduction  of  a  para-magnetic  body  into  a  previously 
uniform  field  of  force  disturbs  the  uniformity  of  the  field.  The 
lines  of  force,  which  were  originally  parallel  and  equidistant,  close 
in  upon  the  body  and  become  continuous  with  the  lines  of  induction 
in  its  interior. 


MAGNETISM.  449 

352.  Permeability  and  Susceptibility.  —  That  property  of  a  sub- 
stance in  virtue  of  which  the  lines  of  induction  are  more  or  less 
closely  arranged  than  are  the  lines  of  force  in  the  originally  undis- 
turbed field  is  called  the  Permeability  .of  the  substance.  We  have 

^E+l 

i  ****• 

which  may  be  written  in  the  form 


In  this  equation,  the  quantity  ft  represents  the  permeability,  and  & 
represents  the  Susceptibility.  The  permeability  is  therefore  the 
ratio  of  the  induction  to  the  force  within  the  substance  of  the 
magnetic  body,  while  the  susceptibility  is  the  ratio  of  the  magnetisa- 
tion to  the  magnetising  force,  and  is  a  measure  of  the  readiness  of 
the  body  to  acquire  magnetisation. 

In  a  paramagnetic  substance,  as  we  have  already  seen,  the  lines 
of  induction  are  more  closely  arranged  than  are  the  lines  of  force. 
That  is  to  say,  the  permeability  of  such  a  substance  is  greater  than 
unity  ;  and  therefore  the  susceptibility  is  positive.  On  the  other 
hand,  in  a  diamagnetic  substance,  the  lines  of  induction  are  less 
closely  arranged  than  are  the  lines  of  force  ;  and  so  the  permeability 


FIG.  196. 

is  less  than  unity,  and  the  susceptibility  is  negative.  In  a  para- 
magnetic body,  north  magnetism  is  manifested  at  that  extremity 
which  faces  in  the  direction  in  which  the  external  lines  of  force  are 
drawn;  in  a  diamagnetic  body,  north  magnetism  appears  at  the 
opposite  end.  Consequently,  while  a  paramagnetic  substance  is 
attracted  towards  the  pole  of  a  magnet,  a  diamagnetic  substance  is 
repelled  from  it.  In  Fig.  196,  the  body  marked  p  is  paramagnetic, 
the  body  marked  d  is  diamagnetic. 

More  generally ;  a  paramagnetic  substance  moves  from  weak 
parts  to  strong  parts  of  a  field  of  force,  while  a  diamagnetic  sub- 
stance moves  from  strong  parts  to  weak  parts. 

29 


450 


A   MANUAL    OF    PHYSICS. 


353.  Residual  Magnetism.     Betentiveness.     Coercive  Force. — 
We  have  already  seen  that  some  substances,  such  as  steel,  retain  to 
a  considerable  extent  their  state  of  magnetisation  after  the  magne- 
tising force  is  removed.     The  property  in  virtue  of  which  this  occurs 
is  termed  Betentiveness. 

The  magnetism  which  remains,  because  of  retentiveness,  is 
called  Besidual  Magnetism.  From  its  great  retentiveness,  hard 
steel  is  employed  in  the  construction  of  so-called  permanent  magnets. 
The  residual  magnetism  of  a  long  bar  of  steel  is  more  permanent 
than  is  that  of  a  short  bar,  for  the  self -demagnetising  force  (§  351) 
has  less  influence  in  the  former  case  than  it  has  in  the  latter. 

Betentiveness  has  very  different  values  in  different  materials.  It 
is  relatively  small  in  good  specimens  of  soft  iron. 

In  order  to  get  rid  of  residual  magnetism  in  any  substance,  we 
must  either  heat  the  substance  to  redness,  or  employ  a  reverse 
magnetising  force.  It  is  usual,  therefore,  to  speak  of  a  Coercive 
Force  as  existent  in  the  material  in  virtue  of  which  residual 
magnetism  is  retained. 

354.  Belation  connecting  Magnetisation  and  Magnetising  Force. 
— If  the  magnitudes  of  any  two  of  the  quantities  IS,  $,  and  ^),  are 
determined  in   any  particular  case,   the   value   of   the   remaining 
quantity  can  be  calculated  from  the  relation  13  =  47r!-}-cf)'     Methods 
for  the  determination  of  each  of  the  three  quantities  will  be  de- 
scribed subsequently. 


FIG.  197. 

Fig.  197  represents  the  usual  course  of  the  variation  of  intensity 
with  magnetising  force.  The  force  is  measured  along  the  axis  O;ij, 
while  the  intensity  of  magnetisation  is  measured  along  the  axis  O1E. 


MAGNETISM.  451 

At  first,  while  the  force  is  small,  the  magnetisation  increases  very 
slowly,  and  at  a  sensibly  uniform  rate.  Then,  as  the  force  is 
increased,  the  law  becomes  $.  =  0$ -\-bffi,  a  and  6  being  constants. 
After  this,  a  very  slight  increase  of  the  magnetising  force  produces 
a  great  change  in  the  magnetisation.  With  still  larger  forces,  the 
rate  of  variation  becomes  rapidly  smaller,  and  ultimately  the 
magnetisation  becomes  sensibly  constant.  These  various  stages  in 
the  process  of  magnetisation  are  represented  by  the  parts  OA,  AB, 
and  BC,  of  the  curve.  If  the  force  be  now  gradually  removed,  the 
magnetisation  will  diminish  at  a  relatively  slow  rate,  until,  when 
the  force  is  entirely  removed,  a  considerable  amount  of  residual 
magnetisation  remains.  This  is  represented  by  OD. 

If  a  reverse  force  be  now  applied,  the  magnetisation  will  fall  off 
rapidly  in  magnitude,  and  will  disappear  entirely  when  the  reverse 
force  has  a  definite  value  OE.  This  may  be  supposed  to  represent, 
as  Hopkinson  suggests,  the  coercive  force. 

If  the  reverse  force  be  now  increased  until  it  reaches  a  value 
equal  to  the  maximum  value  of  the  direct  force,  if  it  be  then 
diminished  to  zero,  and  if,  finally,  positive  force  be  reapplied  until 
the  original  maximum  value  is  attained,  the  magnetisation  will  pass 
through  successive  values  represented  by  the  part  EC'D'E'C. 

The  curve  OR  represents  the  residual  magnetisation  which  is  left 
after  various  magnetising  forces  have  been  applied  and  removed. 

The  dotted  curve  represents  the  change  which  takes  place  in  the 
magnetisation  when  the  same  substance  (say  a  soft  iron  wire)  is 
hardened  by  being  stretched  beyond  its  limits  of  elasticity.  The 
maximum  magnetisation  is  lessened.  The  residual  magnetisation  is 
also  lessened,  but  the  coercive  force  is  increased. 

Since  the  magnetisation  practically  reaches  a  maximum  value 
when  the  force  is  sufficiently  great,  the  substance  is  then  said  to  be 
saturated.  However  much  the  force  may  be  further  increased,  the 
intensity  remains  appreciably  constant. 

The  susceptibility  increases  from  a  small  value  to  a  maximum 
which  is  indicated  by  the  tangent  drawn  from  0  to  the  curve  OBC. 
Thereafter  it  diminishes  to  zero  as  the  force  increases  without  limit. 
The  relation  ju  =  47r&+l  shows  that  the  permeability  also  increases 
from  a  small  value  to  a  maximum  (attained  at  a  somewhat  greater 
value  of  35  than  that  at  which  the  maximum  susceptibility  is 
reached),  after  which  it  gradually  diminishes  to  unity  as  the  force  is 
indefinitely  increased. 

When  soft  iron  is  magnetised  (more  especially  when  the  force  is 
feeble  and, the  specimen  of  iron  is  large)  it  is  found  that  the 
magnetisation  takes  some  time  to  attain  its  full  value  corresponding 

29—2 


452  A   MANUAL   OF   PHYSICS. 

to  the  force  which  is  acting.  This  effect  is,  by  analogy,  said  to  be 
due  to  Magnetic  Viscosity. 

355.  Hysteresis. — We  see  from  Fig.  197,  that  the  changes  of 
magnetisation  tend  to  lag  behind  the  changes  of  force  which  give 
rise  to  them.  Thus,  when  the  stage  C  has  been  reached,  a  much 
greater  change  in  the  value  of  the  force  is  requisite  in  order  to  effect 
a  given  diminution  in  the  magnetisation  than  was  requisite  for  the 
production  of  an  equal  increase  of  magnetisation  just  before  the 
stage  C  was  reached.  A  similar  effect  is  observable  at  the  point 
C'.  Ewing  has  called  this  tendency  Magnetic  Hysteresis. 

As  the  result  of  hysteresis,  different  values  of  the  magnetisation 
may  correspond  to  one  given  value  of  the  magnetic  force,  and  we 
must  therefore  limit  our  definitions  of  permeability  and  suscepti- 
bility to  the  case  of  a  substance  which  is  originally  unmagnetised, 
and  which  is  subjected  to  a  force  which  increases  in  magnitude  con- 
tinuously from  zero  upwards. 

If  E  represent  the  magnetic  energy  of  the  magnetised  body,  the 
increase  of  energy  which  accompanies  an  increase  of  intensity  of 

magnetisation  d$.  is  — -  <?£.     Now  §  62,  d~EldZ  is  the  force  which 
an 

produces  the  change  d$  :  that  is,  it  is  the  force  H.  Hence  the 
increment  of  energy  per  unit  of  volume  which  accompanies  the 
increment  of  dJE  is  HdE;  and  therefore  (compare  §  34)  the  area 
CDC'D'C  (Fig.  197)  represents  an  amount  of  energy  which  has  been 
transformed,  per  unit  of  volume,  in  the  given  cyclical  process.  This 
energy  takes  the  form  of  heat,  and  is  dissipated.  Consequently, 
rapid  reversals  of  magnetisation  will  cause  the  temperature  of  the 
magnetised  substance  to  increase  markedly;  and  no  amount  of 
lamination,  such  as  is  used  in  transformers  or  in  the  armature 
cores  of  dynamos  for  the  prevention  of  heating  by  induced  currents, 
will  prevent  this  effect. 

No  dissipation  of  energy  occurs  if  the  cyclical  changes  in  the 
magnetising  force  are  small,  and  take  place  either  very  rapidly  or 
very  slowly.  For,  in  the  former  case,  no  time  is  allowed  for  a  dimi- 
nution to  occur  in  the  amount  of  lag  of  the  magnetic  effect  behind 
the  change  of  force  which  produces  it,  whether  in  the  direct  or  in 
the  reverse  part  of  the  cycle ;  so  that  the  direct  changes  of  magneti- 
sation are  exactly  reversed  when  the  force  is  reversed  ;  and,  in  the 
latter  case,  complete  time  is  allowed  to  prevent  noticeable  lag  from 
making  its  appearance,  that  is  to  say,  the  changes  in  the  force  take 
place  so  slowly  that  the  proper  changes  of  magnetisation  can  ensue  at 
all  stages  of  the  process ;  and  so,  again,  the  reverse  changes  of  mag- 
netisation follow,  in  the  opposite  direction,  the  same  course  as  the 


MAGNETISM. 


453 


direct  changes.  In  any  other  case  dissipation  of  energy  will  be 
manifested. 

It  appears,  therefore,  that  the  so-called  magnetic  viscosity  tends 
to  produce  hysteresis.  But,  though  this  is  so,  the  converse  state- 
ment that  the  existence  of  hysteresis  implies  the  existence  of  viscosity 
is  neither  necessarily  nor  actually  true. 

356.  Effects  of  Vibration  and  of  Temperature. — Vibration  has  a 
very  great  influence  upon  the  susceptibility  of  a  magnetised  body. 
This  effect  is  very  marked  when  the  magnetising  force  is  small,  but 
is  not  very  noticeable,  if  at  all,  when  powerful  forces  are  used.  It 
increases  the  susceptibility  of  the  substance,  but  diminishes  residual 


FIG.  198. 

magnetism,  coercive  force,  and  hysteresis.  These  results  are 
exhibited  in  Fig.  198,  in  which  the  full  curve  represents  a  cycle 
performed  under  the  condition  of  no  vibration ;  while  the  dotted 
curve  represents  the  result  of  an  experiment  made  upon  the  same 
substance  under  similar  conditions  of  force — the  substances,  how- 
ever, being  tapped  after  each  change  in  the  magnitude  of  the  force. 

The  temperature  of  a  magnetic  substance,  too,  has  a  very  marked 
effect  upon  its  susceptibility.  In  iron,  cobalt,  and  nickel,  increase 
of  temperature  (from  ordinary  values)  first  increases  the  suscep- 
tibility, and  afterwards  diminishes  it,  as  the  magnetising  force  is 
continuously  increased ;  and  the  magnetic  properties  entirely,  and 
suddenly,  vanish  when  the  temperature  attains  a  certain  value 
which  is  different  for  each  substance,  and  varies  to  some  extent  also 
from  one  to  another  specimen  of  any  substance. 

The  temperature  at  which  the  susceptibility  vanishes  is  called  the 
Critical  Temperature.  It  is  a  temperature  at  which  some  fun- 


454  A   MANUAL    OF    PHYSICS. 

damental  change  takes  place  in  the  physical  constitution  of  the 
metal.  The  electric  resistance  of  this  metal  changes  suddenly  at  this 
point,  as  also  does  its  thermo-electric  power  (§  328).  When  the 
reverse  change — from  the  non-magnetic  condition  to  the  magnetic 
condition — takes  place,  as  hard  steel  is  cooled  down  from  a  tem- 
perature higher  than  the  critical  temperature,  a  sudden  liberation  of 
heat  takes  place,  and  the  metal  glows  brightly,  although  it  had  pre- 
viously cooled  to  dull  redness. 

In  iron,  the  suddenness  with  which  the  magnetisation  is  lost  as 
the  critical  temperature  is  approached,  depends  very  largely  upon 
the  value  of  the  magnetising  force.  When  the  force  is  very  small, 
the  susceptibility  first  increases  with  extreme  rapidity  to  a  maximum, 
and  then  diminishes  with  even  greater  rapidity,  as  the  critical  tem- 
perature is  approached.  With  higher  forces,  the  variation  becomes 
much  less  marked. 

There  is  little  or  no  evidence  of  hysteresis  with  regard  to  the 
magnetic  effects  which  follow  changes  of  temperature  unless  the 
critical  temperature  be  included  within  the  cyclical  range.  But  it 
at  once  becomes  evident  when  the  range  includes  the  critical  tem- 
perature ;  for  the  temperature  at  which  the  magnetic  effects  re- 
appear as  the  temperature  is  reduced,  is  lower  than  the  critical 
temperature  at  which  they  disappear  when  the  temperature  is 
raised. 

This  lag  of  magnetic  effect  behind  the  change  of  temperature 
which  gives  rise  to  it,  is  abnormally  evident  in  certain  alloys  of 
nickel  and  iron.  An  alloy  containing  25  per  cent,  of  nickel  was 
found  by  Hopkinson  to  lose  its  magnetic  properties  at  a  temperature 
of  580°  C.,  and  to  remain  non-magnetic  until  its  temperature  fell 
somewhat  below  the  Centigrade  zero.  This  fact  suggests  the  idea 
that  non-magnetic  manganese  steel  may  become  magnetic  if  its 
temperature  be  sufficiently  reduced,  and  that  possibly  all  the  non- 
magnetic metals  may  act  similarly. 

357.  Effects  of  Stress. — Alteration  of  the  state  of  stress  to  which 
the  magnetic  metals  are  subjected,  produces  considerable  alteration 
of  the  magnetic  qualities  of  the  metals. 

Matteuci  observed  that  extension  of  an  iron  rod  produced  an  in- 
crease of  magnetisation,  and  Villari  found  that  when  the  field  is 
sufficiently  intense,  extension  causes  decrease  of  magnetisation. 
This  effect  is  called  the  '  Villari  reversal.' 

The  various  effects  of  longitudinal  and  of  torsional  stress  have 
been  very  fully  investigated  by  Wiedemann,  Sir  W.  Thomson,  and 
others. 

Compression  of  an  iron  rod  produces  effects  opposite  to  those 


MAGNETISM.  455 

which  are  produced  by  extension.  Compression  and  extension, 
respectively,  of  nickel  and  cobalt  rods  also  produce  respectively 
opposite  effects,  but  there  is  no  Villari  reversal  in  this  case ;  for 
all  values  of  the  magnetising  force,  extension  produces  diminution, 
and  compression  produces  increase,  of  the  magnetisation.  The 
diminution  of  the  residual  magnetisation  of  nickel,  under  extending 
stress,  is  even  more  evident  than  is  the  diminution  of  induced  mag- 
netisation. Hysteresis,  under  cyclical  variation  of  load,  is  much 
more  marked  in  the  case  of  iron  than  it  is  in  the  case  of  nickel. 

From  the  above  result  regarding  the  effect  of  extension  on  the 
magnetisation  of  an  iron  rod  in  a  weak  field,  we  can,  by  a  double 
application  of  the  principle  of  stable  equilibrium  (§  15),  deduce  the 
result  that,  in  weak  fields,  increase  of  magnetisation  causes  increase 
of  length ;  or,  conversely,  we  can  deduce  the  former  result  from  the 
latter.  Magnetic  energy  enters  the  rod  from  the  external  medium, 
and,  in  part,  is  transformed  into  potential  energy  of  molecular  con- 
figuration within  the  rod;  and  this  potential  energy  may  be,  in 
turn,  transformed  into  external  work  as  the  length  of  the  rod  alters. 
Let  us  suppose,  first,  that  the  length  of  the  rod  is  not  allowed  to 
alter.  Increase  of  magnetisation  will  then  give  rise  to  pressure  on 
the  restraining  surfaces  which  prevent  alteration  of  length.  Con- 
versely, by  the  principle  of  stable  equilibrium,  decrease  of  pressure 
will  cause  an  increase  of  magnetisation  under  the  given  external 
magnetic  force.  But,  again,  increase  of  pressure  upon  the  restrain- 
ing surfaces  will  result  in  increase  of  length  of  the  rod  if  the  re- 
straint be  removed.  Conversely,  increase  of  length  will  cause  a 
diminution  of  pressure.  We  may  represent  these  results  by  the 
symbolical  expression 

+M  »-»  +P  »-»  +L 

II 
+M  <-«  -P<-is  +L 

where  M,  P,  and  L  represent  respectively  magnetisation  in  weak 
fields,  pressure,  and  length.  Disregarding  the  intermediate  step, 
the  symbols  state  that  increase  of  the  magnetisation  of  an  iron  rod 
in  weak  fields  causes  increase  of  length,  and  that  increase  of  length 
of  the  rod  induces  an  increase  of  magnetisation.  In  strong  fields 
the  expression  would  become 

->  -Ps»-»  -L 

II 
H-M  <-ss  +P  <-«  -L 

where  —  P  may  be  translated  '  increase  of  tension.' 


456 


A    MANUAL    OF    PHYSICS. 


[It  is  important  to  observe  that,  when  we  regard  only  changes  of 
magnetisation  and  of  pressure  (or  tension),  we  are  dealing  with 
energy  flowing  from  an  external  system  into  the  iron ;  and  that, 
when  we  regard  only  changes  of  pressure  (or  tension)  and  of  length, 
we  are  dealing  with  energy  flowing  from  the  iron  rod  to  another 
external  system  ;  while,  when  we  regard  changes  of  magnetisation 
and  of  length  alone,  we  are  dealing  with  energy  flowing  through 
the  rod  from  one  external  system  to  another,  and  being  transformed 
through  its  agency  from  one  form  to  another.] 

Joule  proved  that  no  observable  change  of  volume  takes  place 
when  a  rod  of  iron  is  magnetised,  and  therefore  that  longitudinal 
magnetisation  in  weak  fields  must  cause  a  diminution  of  the  sec- 
tional area  of  the  rod.  Hence  he  concluded  that  if  a  rod  of  iron  be 
magnetised  circularly,  that  is,  if  the  lines  of  magnetisation  be  circles 
surrounding  the  axis  of  the  rod,  longitudinal  contraction  will  ensue. 
He  verified  this  conclusion  by  experiment. 

Torsional  strain,  also,  is  accompanied  by  variations  in  the  magnetic 
qualities  of  iron,  nickel,  and  cobalt  rods.  These  effects  can,  as  Sir 
W.  Thomson  has  shown,  be  deduced  from  the  known  effects  of 
longitudinal  stress  upon  the  magnetic  qualities.  Thus  it  is  known 
that  the  susceptibility  of  iron  in  weak  fields  is  increased  along  lines 


FIG.  199. 


of  traction,  and  is  decreased  along  lines  of  compression.  But,  when 
a  circular  rod  of  iron  is  twisted  in  the  manner  which  is  indicated  by 
the  arrows  in  Fig.  199,  all  lines  such  as  a  a1  suffer  traction,  while 
all  lines  such  as  b  b'  suffer  compression.  Hence  the  susceptibility  is 
increased  along  a  a'  and  is  diminished  along  b  b'.  The  effect  of  this 
is  practically  to  produce  two  components  of  magnetisation — one 
longitudinal,  the  other  circular — when  the  twist  is  sufficiently  great. 


MAGNETISM.  457 

Hence  torsion  in  weak  fields  diminishes  the  longitudinal  susceptibility 
of  iron. 

Conversely,  a  circularly  magnetised  iron  rod,  when  twisted, 
becomes  longitudinally  magnetised.  It  is  easy  to  deduce  the  cor- 
responding effects  in  nickel  and  cobalt.  [No  reversal  of  the  direc- 
tion of  longitudinal  magnetisation  takes  place  in  iron,  however 
strong  the  circular  magnetisation  may  be.  Ewing  explains  this  by 
the  fact  that  the  intensity  of  magnetisation  in  the  direction  of  the 
line  of  traction,  or  of  compression,  never  reaches  the  point  at  which 
the  Villari  reversal  occurs.] 

Since  torsional  stress  produces  circular  magnetisation  in  a  longi- 
tudinally magnetised  rod,  and  since  it  also  gives  rise  to  longi- 
tudinal magnetisation  in  a  circularly  magnetised  rod,  we  might 
expect  that  the  superposition  of  longitudinal  and  circular  magneti- 
sations would  cause  torsional  strain.  This  effect  was  discovered 
experimentally  by  Wiedemann  in  the  case  of  iron.  The  twist  in 
weak  fields  takes  place  in  such  a  direction  as  to  be  completely 
explainable  by  the  increase  of  length  which  occurs  in  the  direc- 
tion of  resultant  magnetisation.  Knott  has  shown  that  the  twist 
occurs  in  the  reverse  way  in  nickel — a  result  which  he  expected 
to  find,  since  nickel  contracts  in  the  direction  of  magnetisation. 

We  may  observe  here  that  the  twisting  of  a  magnetised'  rod,  or 
the  magnetisation  of  a  twisted  rod,  gives  rise  to  a  transient  electric 
current  in  the  magnetised  material.  This  effect  will  be  considered 
in  next  chapter. 

358.  Magnetometric  Measurements. — The  magnetometer  consists 
essentially  of  a  small  magnet,  which  is  suspended  by  a  long  fine 
fibre,  whose  co-efficient  of  torsion  is  negligeable  in'most  cases,  and 
which  is  free  to  turn  about  that  fibre  as  an  axis.  A  small  mirror 
is  usually  attached  to  the  magnet,  so  that,  by  means  of  a  reflected 
beam  of  light,  very  small  angular  motions  of  the  magnet  may  be 
made  evident.  This  apparatus  is  placed  in  a  uniform  field  of  force 
of  known  intensity,  say  the  earth's  field  (§  359).  The  magnet 
then  places  its  length  in  the  direction  of  the  force  in  the  given 
field. 

Let  the  magnet  be  placed  at  P  (Fig.  200),  and  let  the  direction  of 
the  controlling  force  be  PQ.  Let  AB  be  a  bar  magnet,  the  intensity 
of  magnetisation  of  which  we  have  to  determine,  and  let  the  points 
A  and  B  represent  the  position  of  its  poles.  Place  it  symmetrically 
with  regard  to  PQ  in  the  position  which  is  indicated  in  the  figure. 
Let  I  be  its  (unknown)  intensity  of  magnetisation,  while  a  is  its 
sectional  area.  Then  la  is  the  strength  of  its  pole.  The  effect  of 
the  north  pole,  A,  at  P  is  in  the  direction  AP,  and  is  equal  to 


458 


A  MANUAL  OF  PHYSICS. 


I&/AP2.  Similarly,  the  directive  force  of  the  south  pole,  B,  at  P  is 
in  the  direction  PB,  and  is  equal  to  —  Ia/PB2.  If  we  represent 
these  forces  by  AP  and  PB  respectively,  it  is  obvious  that  the  re- 
sultant effect  of  the  two  is  represented  on  the  same  scale  by  AB. 
The  magnitude  of  the  resultant  is  therefore  IaAB/AP3,  and  acts  so 
as  to  place  the  little  magnet  at  P  parallel  to  AB,  with  its  poles 
facing  oppositely  to  those  of  AB. 


B 


F  G.  200. 


Now  let  PR  represent  this  force  on  the  same  scale  that  PQ  repre- 
sents the  force  of  the  external  field.  PS  is  the  resultant  of  these 
forces,  and  the  little  magnet  sets  itself  in  the  direction  PS,  making 
an  angle  9  with  PQ,  such  that  tan  0  =  PR/PQ.  If  F  be  the  intensity 
of  the  external  "force,  this  gives 


(1) 


from  which  we  can  calculate  I. 

In  order  to  find  the  value  of  the  force  F,  if  it  be  unknown,  we 
may  set  the  magnet  AB  oscillating  under  the  action  of  F  alone. 
The  time,  T,  of  a  small  oscillation  is  given  by  the  equation 


T2~~  ~  K     ' 

where  K«is  the  moment  of  inertia  of  the  magnet  about  its  axis 
of  suspension,  and  AB  is  its  length.  For  if  the  direction  of  the 
force  F  be  denoted  by  the  arrow  (Fig.  201),  and  if  ns  represent  the 
magnet  inclined  at  an  angle  9  to  the  direction  of  the  force,  FS  in 
9  is  the  force  which  is  acting  perpendicularly  to  the  length  of  the 
magnet,  and  which  tends  to  turn  it  around  its  axis  of  suspension. 
This  force  acts  at  each  end  of  the  magnet  so  as  to  produce  rotation 


MAGNETISM. 


459 


in  the  positive  direction,  and  the  turning  moment  is  therefore 
F  sin  9  laAB.  When  the  angle  is  small,  this  becomes  F0I&AB.  The 
angular  acceleration  is  0  (§§  42,  45),  and  the  momentum  which  is 
porduced  per  unit  of  time  is  m0K,  where  m  is  the  mass  of  the  magnet 


/s 


n> 

F 

FIG.  201. 

and  E  is  its  radius  of  gyration  (§  70).  Hence  the  moment  of 
momentum  which  is  produced  per  unit  of  time  is  ra0B2=K0,  where 
K  is  the  moment  of  inertia  (§  70).  We  therefore  have 


the  minus  sign  being  used,  since  the  angular  acceleration  is  nega- 
tive. Now  every  quantity  in  this  equation  is  constant,  with  the 
exception  of  0  ;  and  so  the  equation  expresses  the  fact  that  the 
angular  acceleration  is  negative  and  is  proportional  to  the  displace- 
ment. The  small  oscillations  of  the  magnet,  therefore,  obey  the 
simple  harmonic  law,  and  the  angular  position  of  the  magnet  is 
given  (§  51)  by  the  equation 


0— Pcos 


where"  P  and  Q  are  constants  and  t  is  the  time  ;  whence  T  being  the 
periodic  time,  we  get  (§  51) 


By  elimination  between  the  equations  (1)  and  (2),  we  can  find  the 
values  of  F  and  of  laAB  (which  is  the  magnetic  moment  of  the 
magnet).  Also,  a  and  AB  being  known,  we  can  obtain  the  value  of 
I  ;  and  hence,  if  we  know  the  intensity  of  the  magnetising  force, 
we  can  calculate  the  susceptibility  and  the  permeability  of  the 
substance. 

Equation  (2)  shows  that  the  intensities  of  two  fields  are  inversely 


460  A   MANUAL    OF   PHYSICS. 

proportional  to  the  squares  of  the  periods  of  the  small  oscillations 
of  a  magnet  of  known  magnetic  moment,  which  is  suspended  first 
in  one  field  and  then  in  the  other. 

The  ballistic  method  of  making  magnetic  measurements  will  be 
discussed  in  §  369. 

359.  Terrestrial  Magnetism. — The  earth  exerts  a  magnetic 
action  in  virtue  of  which  compass  needles  point  in  a  northerly 
direction.  The  angular  distance  between  the  line  along  which  a 
compass  needle  points  and  the  geographical  meridian  is  called  the 
magnetic  declination  or  variation.  The  magnetic  needle,  if  it  were 
carefully  supported  on  an  axis  which  passes  through  its  centre  of 
inertia,  and  which  is  perpendicular  to  the  magnetic  meridian,  would, 
in  Britain,  place  its  magnetic  axis  in  a  direction  which  is  inclined  to 
the  horizon — the  north  end  pointing  downwards  at  a  considerable 
angle.  This  angle  is  called  the  Magnetic  Dip. 

The  declination  and  the  dip  vary  considerably  from  one  part  of 
the  earth's  surface  to  another.  In  some  regions  the  declination  is 
easterly,  in  others  it  is  westerly.  The  line  on  the  earth's  surface,  at 
all  points  of  which  the  declination  is  zero,  is  called  the  Magnetic 
Equator.  It  does  not  coincide  with  the  geographical  equator,  and 
is  not  a  great  circle.  That  point  on  the  surface  at  which  the  north 
pole  of  a  magnet  points  vertically  downwards  is  called  the  North 
Magnetic  Pole  of  the  earth,  and  that  point  at  which  the  south  pole  of 
a  magnet  points  vertically  downwards  is  called  the  South  Magnetic 
Poleoi  the  earth.  These  poles  do  not  coincide  with  the  geographical 
poles  of  the  earth,  neither  do  they  lie  at  opposite  extremities  of  a 
diameter.  [Observe  that  the  magnetism  which  we  may  suppose  to 
be  collected  at  the  north  pole  of  the  earth  must  be  south  magnetism, 
i.e.,  it  must  be  of  the  same  kind  as  that  which  appears  at  the  south- 
pointing  pole  of  a  magnet.  Similarly,  the  magnetism  which  is 
manifested  at  the  south  magnetic  pole  of  the  earth  must  be  north 
magnetism.] 

The  earth's  magnetic  force  is  in  a  constant  state  of  variation.  It 
changes  with  the  hour  of  the  day  and  the  time  of  the  year ;  and  it 
depends  also  upon  the  position  of  the  moon.  Yet  these  variations 
do  not  appear  to  be  due  to  any  direct  action  of  the  sun  or  the 
moon. 

Sudden  disturbances  sometimes  take  place  in  addition  to  these 
more  regular  variations.  A  period  of  maximum  disturbance 
occurs  every  eleven  years,  and  coincides  with  the  period  of  maxi- 
mum sun-spot  disturbance. 

A  slow  secular  change  of  the  position  of  the  magnetic  poles  is 
also  in  progress. 


MAGNETISM.  461 

360.  Theories  of  Magnetism. — At  one  time  magnetic  phenomena 
were  explained  by  the  assumption  of  the  existence  of  two  imponder- 
able fluids,  one  of  which  constituted  north  magnetism,  while  the 
other  constituted  south  magnetism.  In  Poisson's  elaboration  of  this 
theory  a  magnetic  body  was  supposed  to  be  made  up  of  spheres  of 
infinite'' permeability,  uniformly  distributed  in  an  absolutely  non- 
permeable  fluid.  This  made  the  problem  of  magnetic  induction 
identical  with  that  of  electric  induction  in  a  non-conducting 
medium,  throughout  which  perfectly  conducting  insulated  spheres 
were  uniformly  distributed.  Among  other  objections  to  this  theory 
is  the  fatal  one  pointed  out  by  Maxwell,  that  the  permeability  of 
iron  is  too  great  to  be  accounted  for  even  if  the  spheres  were  packed 
in  the  closest  possible  arrangement. 

In  modern  theories  the  molecules  are  supposed  to  be  little 
magnets.  In  an  unmagnetised  body  the  magnetic  molecules  have 
their  axes  distributed,  on  the  whole,  uniformly  in  all  directions ;  and 
the  substance  becomes  magnetised  when  the  axes  of  its  molecules 
get,  on  the  whole,  a  definite  set  in  one  direction.  Saturation  will 
take  place  when  all  the  molecules  have  set  their  axes  in  the  direc- 
tion of  the  magnetising  force. 

The  fact  that  the  slightest  force  does  not  produce  saturation 
shows  that  displacement  of  the  molecules  must  be  resisted  by  some 
force.  Weber  assumed  that  each  molecule  is  acted  upon  by  a  constant 
force  in  the  original  direction  of  its  axis,  which  tends  to  prevent  its 
orientation  from  that  direction,  and  tends  to  make  it  resume  its 
original  direction  when  it  is  displaced  from  it.  It  follows  from  this 
assumption  that  the  curve  of  magnetisation  (Fig.  197)  should  at  first 
be  a  straight  line,  that  it  should  afterwards  become  concave  to  the 
axis  along  which  the  force  is  measured,  and  that  it  should  ultimately 
approach  an  asymptote  parallel  to  that  axis.  This  does  not  agree 
with  experiment. 

Maxwell  improved  this  hypothesis  by  the  additional  assumption 
that  a  molecule  could  return  to  its  original  position  if  it  were 
turned  through  an  angle  of  less  than  a  certain  finite  magnitude,  and 
that  if  it  were  displaced  through  an  angle  greater  than  this,  it  would 
retain,  after  removal  of  the  force,  a  displacement  equal  to  the 
excess  of  its  total  displacement  over  this  quantity.  This  form  of 
the  theory  leads  to  a  magnetisation  curve  similar  to  that  given  by 
Weber's  unmodified  theory,  and  it  indicates  that  the  curve  of 
residual  magnetisation  starts  from  a  point  on  the  force-axis  at  a 
finite  distance  from  the  origin,  is  always  concave  to  that  axis,  and 
approaches  a  parallel  asymptote.  These  results  also  are  incorrect. 

Ewing,  following  a  limit  by  Maxwell,  regards  each  molecule  as 


462  A    MANUAL    OF    PHYSICS. 

subject  only  to  the  mutual  action  of  the  entire  system  of  surround- 
ing molecules.  He  has  constructed  a  model  of  such  a  system  by 
means  of  a  number  of  pivoted  magnets,  which  are  arranged  in 
parallel  rows.  So  long  as  no  external  magnetic  force  acts,  the 
magnets  arrange  themselves  in  positions  of  stable  equilibrium  under 
their  mutual  forces,  some  of  them  pointing  in  one  direction,  some 
in  another.  This  illustrates  the  condition  of  non-magnetised  steel. 
If  only  a  feeble  uniform  magnetic  force  acts,  each  magnet  is  slightly 
turned  from  its  first  position,  which  it  reassumes  when  the  force  is 
removed.  This  illustrates  the  first  stage  in  the  process  of  magneti- 
sation. A  somewhat  stronger  force  causes  instability  in  the  origin- 
ally less  stable  groups  of  the  magnets,  and  the  magnets  which 
compose  these  groups  swing  round  into  a  new  stable  position.  As 
the  external  force  is  increased  still  further,  more  and  more  groups 
break  up,  until  all  have  taken  the  new  position  of  equilibrium  under 
their  own  mutual  forces  and  the  external  directive  force.  This 
illustrates  the  second  stage  of  magnetisation,  in  which  the  ratio  of 
magnetisation  to  magnetising  force  increases  with  great  rapidity. 
The  third  stage,  in  which  this  ratio  is  practically  constant,  is 
exemplified  by  the  fact  that  an  infinite  force  is  now  needed  to  make 
the  magnets  point  exactly  in  the  direction  of  the  external  lines  of 
force.  If  the  external  force  be  now  removed,  a  considerable  pro- 
portion of  the  magnets  retain  their  final  positions  of  equilibrium — 
in  other  words,  magnetic  retentiveness  is  exhibited. 

This  model  can  also  show  the  effects  of  strain  on  the  magnetic 
properties.  For  this  purpose  the  magnets  are  placed  on  a  sheet  of 
indiarubber.  If  the  indiarubber  is  stretched  the  magnets  are 
separated  out  from  one  another  in  one  direction,  and  are  brought 
nearer  to  each  other'  in  a  direction  at  right  angles  to  the  former. 
The  magnetic  susceptibility  is  increased  or  is  diminished,  according 
as  the  stability  of  the  magnets  is  diminished  or  is  increased  by  the 
alteration  of  relative  position.  Similarly,  the  increase  of  the 
susceptibility  of  iron  with  rise  of  temperature  is  explained  by 
the  diminution  of  mutual  magnetic  influence  which  results  from 
increased  distance.  Professor  Ewing  suggests  that  the  total  loss  of 
magnetisation  which  occurs  at  a  high  temperature  is  due  to  a  con- 
tinuous whirling  motion  of  the  magnetic  molecules.  He  suggests, 
also,  that  the  dissipation  of  energy,  which  occurs  when  hysteresis  is 
exhibited,  is  due  to  the  induced  electric  currents  (§  342),  which  are 
caused  by  angular  motions  of  the  magnetic  molecules. 


CHAPTER  XXXI. 

ELECTROMAGNETISM,    ETC. 

361.  Oersted's  Discovery. — Oersted  found  that  a  magnetic  needle 
always  tends  to  place  its  length  in  a  direction  at  right  angles  to  a 
plane  which,  passing  through  its  centre,  contains  a  linear  circuit, 
through  which  an  electric  current  is  flowing.  The  direction  in 
which  the  north  pole  points  depends  upon  the  direction  in  which 
the  current  flows.  The  north  pole  always  tends  to  move  round  the 
linear  circuit  in  a  direction  which  is  related  to  that  of  the  current  in 
the  way  in  which  the  rotation  of  a  right-handed  screw  is  related  to 
its  linear  motion. 

After  this  fact  was  discovered,  it  was  surmised  that  the  converse 
phenomenon  might  also  be  found  to  exist — that  motion  of  the  linear 
circuit,  through  which  the  current  flows,  would  take  place  if  the 
magnet  were  fixed  while  the  circuit  was  free  to  move.  This  was 
verified  experimentally  by  Ampere. 

And,  further,  since  a  magnet  can  act  thus  upon  two*  neighbouring 
circuits,  through  which  electric  currents  flow,  it  was  supposed  that 
mutual  action  might  be  found  to  exist  between  these  circuits  them- 
selves if  the  magnet  were  removed.  Ampere  proved  the  existence 
of  this  action  also. 

362.  Magnetic  Action  of  Closed  Electric  Circuits. — The  above 
statement  regarding  Oersted's  discovery  shows  that  a  linear  electric 
current  is  surrounded  by  circular  magnetic  lines  of  force.  The 
researches  of  Ampere  and  of  Weber  have  enabled  us  to  find  the 
distribution  of  electric  force  in  the  neighbourhood  of  any  conducting 
circuit. 

It  is  found  that  a  small  plane  closed  circuit  produces  the  same 
magnetic  action  as  a  small  magnet,  placed  at  some  point  inside  the 
circuit  with  its  length  perpendicular  to  the  plane,  and  having  the 
direction  of  its  axis  (measured  from  south  pole  to  north  pole) 
related  to  the  direction  of  the  circulation  of  the  current  according  to 
the  rule  of  right-handed  screwing  motion,  while  its  magnetic 


464  A    MANUAL    OF    PHYSICS. 

moment  is  equal  to  the  area  of  the  circuit  multiplied  by  the  strength 
of  the  current.  [The  word  small  means  that  the  point  at  which  the 
action  is  determined  is  very  far  off  in  comparison  with  the  dimen- 
sions of  the  circuit.] 

It  is  of  no  consequence  in  what  part  of  the  circuit  the  equivalent 
magnet  is  placed ;  and  so  we  may  assume  that  it  is  an  extremely 
thin  magnetic  shell,  which  fills  the  entire  circuit,  and  is  possessed 
of  a  magnetic  intensity  which  is  numerically  equal  to  the  strength 
of  the  electric  current  divided  by  the  thickness  of  the  shell.  For,  if 
I,  a,  and  t  represent  respectively  the  magnetic  intensity,  the  area, 
and  the  thickness  of  the  shell,  lat  is  the  moment  of  the  shell,  and, 
therefore,  ia=Iat,  or  i  =  It,  where  i  is  the  strength  of  the  cur- 
rent, and  It  is  called  the  strength  of  the  shell. 

Now,  let  any  finite  circuit  PQES  (Fig.  202)  be  filled  with  a  net- 
work of  infinitely  small  conducting  meshes  of  any  shape.  Let  a 
current  i  circulate  in  the  circuit  in  the  positive  direction.  We  may 
assume  that  an  equal  current  flows  similarly  in  each  of  the  meshes, 
such  &s  ziqrs,  f°r  the  currents  in  each  common  side  of  two  adjacent 


FIG.  202. 

meshes  exactly  neutralise  each  other.  But,  as  above,  the  magnetic 
effect  of  the  current  in  each  mesh  may  be  supposed  to  be  due  to  a 
magnetic  shell  of  strength  i.  And  hence  we  see  that  the  magnetic 
action  of  the  circuit  PQES  at  external  points  is  equivalent  to  that  of 
a  magnetic  shell  of  strength  i,  and  of  any  form,  which  completely 
fills  the  circuit. 

It  is  easy  to  find  a  simple  expression  for  the  magnetic  potential 
of  the  shell.  For,  if  d$  be  an  element  of  its  surface,  we  may  replace 
the  part  of  the  shell  which  corresponds  to  dS  by  a  small  magnet, 


ELECTROMAGNETISM,    ETC.  465 

the  strength  of  whose,  pole  is  idSjt.  The  force  with  which  the  north 
pole  n  (Fig.  203)  acts  on  a  unit  north  pole,  placed  at  a  point  P,  is 
where  r  is  the  distance  from  P  to  n.  The  potential  at  P 


FIG.  203. 

due  to  n  is,  therefore,  idS/tr.  Similarly,  the  potential  of  s  at  P  is 
-  idS[tr'9  where  r'  is  the  distance  from  P  to  s.  The  total  potential 
is,  therefore, 


_ 
:~~ 


since  r'  is  practically  equal  to  r.  But  r'  —  r—t  cos  0,  where  9  is  the 
angle  between  the  axis  of  the  magnet  and  the  line  joining  P  to  its 
centre,  and  t  is  the  length  of  the  magnet  (equal  to  the  thickness  of 
the  shell).  Hence 

idS  cos  0 

Now,  since  the  element  of  the  surface  of  the  shell  is  at  right  angles 
to  the  axis  of  the  magnet,  dS  cos  9  represents  the  resolved  part  of 
the  element  normal  to  r,  and  dS  cos  0/r2  is  the  elementary  solid  angle 
which  the  surface  subtends  at  P.  We  may,  therefore,  write 


wnere  di»  represents  this  elementary  angle.  And,  in  order  to  find 
the  total  potential  V  at  P,  due  to  the  whole  shell,  we  have  merely 
to  sum  all  the  quantities,  such  as  V,  for  the  whole  surface. 
Hence 


that  is,  the  potential  at  any  point  external  to  the  magnetic  shell  is 
equal  to  the  product  of  the  intensity  of  the  current  into  the  solid 
angle  which  the  shell  subtends  at  the  given  point. 

It  follows  that  the  work  which  is  done  upon  a  unit  north  pole  by 
the  magnetic  forces  due  to  the  current  is  ^i-wo),  if  the  pole 
passes  from  a  place  where  w  has  the  value  w  to  a  place  where  it  has 
the  value  w.2-  In  particular,  if  the  pole  completely  describes  a 
closed  path  which  does  not  pass  through  the  interior  of  the  circuit 
in  which  the  current  flows,  the  work  is  zero.  If  the  pole  passes 

30 


466  A   MANUAL    OF    PHYSICS. 

by  an  external  path  from  a  point  P,  which  is  infinitely  close  to  one 
side  of  the  shell,  to  a  point  P',  which  is  infinitely  near  to  P  on  the 
opposite  side  of  the  shell,  w  changes  by  an  amount  which  is  infi- 
nitely nearly  equal  to  47r,  and  the  work  which  is  done  by  the  magnetic 
forces  is  practically  4-n-i.  But  any  shell  of  the  proper  moment, 
which  is  bounded  by  the  circuit,  produces  the  same  magnetic  effect 
at  distant  points  as  the  current  produces;  and  so  we  may  now 
replace  the  shell,  whose  action  we  are  considering,  by  another  shell 
of  the  same  moment,  which  is  everywhere  finitely  distant  from  P 
and  P',  and  which  is  bounded  by  the  same  circuit.  The  forces  due 
to  this  shell  do  infinitely  little  work  upon  the  unit  north  pole  when 
it  passes  from  P'  to  P,  and  hence  the  work  which  is  done  upon  the 
unit  pole  in  a  single  complete  passage  round  a  closed  path  which 
passes  through  the  circuit  is  4?ri.  In  n  such  passages  the  work  is  4-Trin. 
363.  Electrodynamic  Action  on  an  Electric  Circuit.  —  Let  the 
intensity  of  magnetisation  of  a  shell,  which  produces  the  same 
magnetic  action  as  a  given  circuit,  be  I  ;  and  let  the  shell  be  placed 
in  a  field  of  force  the  intensity  of  which  is  F.  We  may  choose  the 
shell  so  that  each  element  of  its  surface  is  at  right  angles  to  the 
direction  of  the  force  in  its  immediate  neighbourhood.  Consider  a 
small  portion  ds  of  the  north  face  of  the  surface.  The  total  force 
acting  on  this  part  is  -\-FIds,  the  plus  sign  being  used  since  we 
consider  it  to  be  positive  when  it  acts  in  the  direction  of  the  outward 
normal  to  the  north  face.  Similarly  the  force  which  acts  on  the 
corresponding  part  of  the  south  face  is  —  FIcZS  ;  and  therefore  the 
potential  energy  of  this  portion  of  the  shell  in  the  given  field  is 
-  FIdSt,  where  t  is  the  thickness  of  the  shell.  Now  it,  the  strength 
of  the  shell  is  equal  to  t,  the  strength  of  the  current  ;  and  so  the 
potential  energy  is  -FidS.  Consequently,  the  total  potential  energy 
of  the  shell  is 

-«N 

where  N  is  the  whole  force  acting  on  the  shell,  that  is,  the  number 
of  lines  of  force  which  pass  through  the  circuit  whose  action  is 
represented  by  that  of  the  shell. 

When  a  force  F  produces  a  change  df  in  one  of  the  quantities 
which  determine  the  position  of  the  circuit,  the  work  which  is 
expended  is  Fdf,  and  the  corresponding  change  in  the  potential 
energy  of  the  circuit  is  idN.  Hence 


And  we  see  that  the  force  F  tends  to  produce  or  to  oppose  the 
change  df  according  as  that  change  is  accompanied  by  an  increase, 


ELECTROMAGNETISM,    ETC.  467 

or  by  a  decrease,  of  the  number  of  lines  of  force  which  pass  through 
the  circuit  in  the  positive  direction. 

In  particular,  if  a  part  of  the  circuit  be  moveable,  the  electro- 
magnetic forces  which  act  upon  the  circuit  will  tend  to  produce  such 
a  displacement  of  the  moveable  part  as  will  cause  an  increase  of  the 
number  of  lines  of  force  which  pass  through  the  circuit. 

864.  Case  of  Linear  Circuits. — We  have  already  seen  that  a 
linear  circuit  carrying  an  electric  current  is  surrounded  by  circular 
lines  of  force,  the  direction  of  which  is  related  to  that  of  the  current 
in  the  same  way  as  the  rotation  of  a  right-handed  screw  is  related 
to  its  linear  motion. 

Let  A  B  (Fig.  204)  represent  part  of  a  fixed  linear  circuit  through 
which  a  current  flows  from  A  to  B,  and  let  ab  be  a  portion  of  a 
moveable  parallel  circuit  through  which  a  current  flows  from  a  to  b. 
We  may  assume  that  the  circuit  ab  is  completed  by  way  of  p.  The 
lines  of  force  due  to  AB  pass  through  abp  in  the  positive  direction, 
and  a  displacement  of  ab  towards  AB  would  increase  the  area  abp, 

B 


(!  J     f 

A  ""*** 

FIG.  204. 

and  so  would  cause  an  increase  in  the  number  of  positively  drawn 
lines  of  force.  Hence  the  electrodynamic  action  between  the 
circuits  is  such  as  to  cause  their  mutual  approach.  [We  would 
arrive  at  the  same  result  by  the  supposition  that  the  circuit  ab  is 
completed  by  way  of  q.  For  the  lines  of  force  due  to  AB  pass 
through  the  electric  circuit  abq  in  the  negative  direction.  Hence 
ab  will  move  so  as  to  diminish  the  area  of  abq,  that  is,  so  as  to 
diminish  the  number  of  negatively  drawn  lines  of  force  which  pass 
through  it.]  Similarly,  mutual  repulsion  will  ensue  if  the  currents 
are  oppositely  directed. 

Next  let  the  circuits  AB  and  ab  be  inclined  to  each  other,  and  let 
00'  (Fig.  205)  be  the  shortest  line  between  them.  Let  us  suppose 
that,  while  AB  is  fixed,  ab  is  free  to  turn  around  00'  as  an  axis. 
Complete  the  circuit  ab  by  way  of  p.  No  lines  of  force  due  to  AB  will 
pass  through  abp  when  AB  and  ab  are  mutually  perpendicular.  On 
the  other  hand,  the  number  of  lines  which  pass  through  it  in  the  posi- 

CO -2 


468 


A    MANUAL    OF    PHYSICS. 


tive  direction  is  a  maximum  when  the  currents  in  AB  and  ab  are 
parallel  and  similarly  directed.    The  electrodynamic  action  on  ab  is 


therefore  such  as  to  cause  it  to  place  itself  parallel  to  AB.  If  either 
current  be  reversed  the  moveable  circuit  will  turn  so  that  ba  is 
co -directional  with  AB. 

365.  Circular  Circuits.  Solenoids.  'Ampere's  Hypothesis  re- 
garding Magnetism. — A  circular  circuit,  of  radius  r,  which  is 
traversed  by  a  current  i,  may  be  replaced  by  its  equivalent  magnetic 
shell  of  intensity  ±i[t  (§  362).  So  also,  a  second  circular  circuit, 
which  is  traversed  by  a  current  i',  may  be  replaced  by  a  shell  of 
intensity  ±i'jtr.  These  shells  will  turn  so  as  to  place  their 
oppositely  magnetised  forces  parallel,  and  will  then  exhibit  mutual 
attraction.  Hence  the  electric  circuits  will  tend  to  turn  so  that  the 
currents  in  each  are  parallel,  and  will  then  exhibit  mutual  attrac- 
tion. But  if,  while  the  circuits  are  in  this  position,  one  of  the 
currents  be  reversed,  mutual  repulsion  will  be  exhibited. 

[It  is  an  easy  matter  to  calculate  the  force  which  such  a  circuit 
carrying  a  current  i,  exerts  at  its  centre.  Let  the  centre  be  at  o 


(Fig.  206)  and  let  a,  c,  be  the  points  in  which  the  circuit  cuts  the 
plane  of  the  paper — the  plane  of  the  circuit  being  supposed  to  be 
perpendicular  to  that  plane.  Imagine  the  circuit  to  be  replaced  by 
an  equivalent  hemispherical  magnetic  shell  (§  362)  abc.  The  work 


ELECTROMAGNETISM,    ETC.  469 

which  is  done  in  displacing  a  unit  pole  from  0  through  a  small 
distance  Od=r  is  i(w  —  <•/),  where  w  and  «'  are  respectively  the 
angles  subtended  at  0  and  d  by  the  shell  abc.  If  r  be  the  radius, 
the  value  of  w  is  27r,  and  the  value  of  a/  is  practically  (27rr2  -  2?rrr)/r2 ; 
so  that  the  work  is  27rir/r.  Hence  the  force,  which  is  prac- 
tically uniform  when  r  is  sufficiently  small,  is  2?ri/r;  so  that, 
when  r  is  unity,  unit  length  of  the  current  exerts  a  force  i  at  the 
centre.] 

These  phenomena  can  be  readily  exhibited  by  means  of  two 
small  floating  cells,  each  of  which  consists  of  a  test  tube  containing 
dilute  sulphuric  acid  into  which  dip  zinc  and  copper  plates  connected 
externally  by  a  circular  copper  wire.  The  test  tubes  are  inserted  in 
pieces  of  cork,  and  are  floated  on  the  surface  ofi  water. 

A  wire  which  is  bent  into  a  cylindrical  helix  in  the  manner  indi- 
cated in  Fig.  207,  is  called  a  Solenoid.  If  it  be  freely  suspended  on 
pivots,  and  be  traversed  by  a  current,  it  will  act  like  a  magnet 
under  the  action  of  the  earth's  force  or  of  other  magnets  or  solenoids. 
If  the  number  of  turns,  n,  per  unit  of  length  is  large,  we  may 


FIG.  207. 

replace  each  nearly  closed  circuit  by  a  shell,  the  intensity  of 
magnetisation  of  which  is  ±ra.  Throughout  the  length  of  the 
solenoid,  the  actions  of  the  shells  are  mutually  annulled,  except  at 
the  ends,  where  quantities  of  magnetism  ±nia  are  found,  a  being'the 
area  of  the  shells.  At  points  which  are  far  distant  in  comparison 
with  the  radius  of  the  solenoid,  the  action  is  therefore  equivalent  to 
that  of  a  magnet  of  moment  nial,  where  I  is  the  length  of  the 
solenoid. 

In  the  interior  of  a  very  long  solenoid,  which  contains  n  turns  per 
unit  of  its  length,  and  through  which  a  current  i  circulates,  the 
total  force  is  equal  to  kirnia,  where  a  is  the  area  of  a  transverse 
section.  For  the  thickness  of  each  shell  equivalent  to  a  turn  of  the 
wire  is  1/w,  and  therefore  the  surface  density  of  the  distribution  of 
magnetism  on  its  two  faces  is  -f  ni  and  —  ni  respectively.  Hence 
the  force  at  any  point  within  each  shell,  and  therefore  throughout 
the  interior  of  the  solenoid,  is  kitni.  Thus  a  solenoid  may  be  used 
for  the  production  of  a  very  intense  magnetic  field;  and  this 


470  A  MANUAL   OF   PHYSICS. 

furnishes  one  of  the  most  convenient  ways  of  temporarily,  or 
permanently,  magnetising  a  magnetic  substance. 

These  properties  of  circular  circuits  and  of  solenoids  led  Ampere 
to  suggest  that  the  molecules  of  magnetic  substances  may  exhibit 
their  magnetic  properties  in  virtue  of  electric  currents  which  circu- 
late within  them  in  closed  circuits. 

366.  Continuous  Rotation  under  Electromagnetic  Force.  Elec- 
tric Motors. — We  have  already  seen  that  no  work  is,  on  the  whole, 
done  upon  a  magnetic  pole  which  describes  a  closed  path  in  a  field 
of  force  due  to  an  electric  circuit,  provided  that  the  path  does  not 
pass  through  the  interior  of  the  circuit.  But  it  is  also  true  that  no 
work  will  on  the  whole  be  done  upon  a  magnetised  body  which 
completely  describes  a  closed  path  passing  through  the  interior  of 
the  circuit ;  for  the  body  is  composed  of  excessively  small  magne- 
tised molecules,  and  the  total  amount  of  work  which  is  expended 
upon  each  molecule  in  the  process  is  zero,  since  its  north  and  south 
poles  are  of  equal  strength. 

But  work  will  be  expended  on  the  whole  if  motion  of  part  of  the 
circuit  takes  place,  without  interruption  of  the  current,  under  the 
action  of  external  magnetic  force.  As  an  example,  let  us  consider 
a  horizontal  circular  conductor  AB  (Fig.  208).  A  current  which 


B 


FIG.  208. 

enters  this  circuit  at  A,  will  divide  into  two  parts,  which  reunite  at 
B  and  flow  through  the  conductor  BC  to  the  point  C,  which  is 
connected  with  the  negative  pole  of  the  battery  to  the  positive  pole 
of  which  the  point  A  is  joined.  The  lines  of  force,  due  to  the  earth's 
magnetic  action,  pass  downwards  through  the  circuit.  In  the 
region  ABC,  their  direction  is  related  to  that  of  the  current  accord- 
ing to  the  law  of  left-handed  screwing  motion  :  in  the  region  to  the 
other  side  of  BC,  their  due  action  is  related  to  that  of  the  current 
according  to  the  law  of  right-handed  screwing  motion.  Hence  the 
electrodynamic  action  upon  BC  will  cause  it  to  rotate  in  the  direc- 


ELECTROMAGNETISM,    ETC. 


471 


tion  of  the  hands  of  a  watch  provided  that  it  is  pivoted  at  C,  and 
has  a  sliding  contact  at  B. 

If  AB  were  a  circular  conducting  disc,  pivoted  at  C,  and  having  a 
sliding  contact  at  its  circumference,  so  that  an  electric  current 
flowed  radially  inwards,  and  if  lines  of  force  (whether  due  to  the 
earth  or  to  external  magnets)  passed  through  it  as  above,  con- 
tinuous rotation  of  the  disc  in  the  direction  of  the  hands  of  a  watch 
would  ensue.  This  arrangement  is  known  as  Barlow's  wheel. 

Conversely,  continuous  motion  of  the  magnet  may  take  place  if 
the  circuit  be  fixed  while  the  magnet  is  free  to  move  and  the 
direction  of  the  current  is  reversed  whenever  the  magnet  passes 
from  one  side  of  the  circuit  to  the  other.  Indeed,  it  is  easy  to 
arrange  the  conditions  in  such  a  way  that  continuous  rotation  of 
the  magnet  will  take  place  without  periodic  reversal  of  the  direction 
of  the  current.  For  example,  let  ns  and  n's'  (Fig.  209)  be  two  magnets, 


B 


D 


n- 


A 
FIG.  209. 

which  are  connected  together  by  the  cross  piece  CD,  and  which, 
being  pivoted  at  B,  are  free  to  rotate  about  AB  ;  and  let  a  current 
flow  continuously  along  AB.  The  direction  of  the  circular  lines  of 
force  which  surround  AB  is  related  to  that  of  the  current  according 
to  the  law  of  right-handed  screwing  motion;  and  therefore  the 
poles  n  and  n'  rotate  around  AB  in  that  direction.  And  it  follows 
that  a  delicately  pivoted  magnet,  along  which  a  current  flows,  will 
be  similarly  set  in  rotation,  for  it  may  be  supposed  to  consist  of  a 
number  of  magnets  grouped  around  its  axis. 

These  principles  of  electrodynamic  action  are  applied  practically 
in  the  construction  of  electro-magnetic  machines  or  motors  for  the 
transformation  of  electric  energy  into  mechanical  work.  The 
electric  circuits  may  be  fixed  while  the  magnets  rotate ;  or,  prefer- 
ably, the  magnets  may  be  fixed  while  the  circuits  rotate. 

867.  Electromagnetic  Induction.  The  'Dynamo.' — We  have  seen 
that  the  increase  of  the  potential  energy  of  an  electric  circuit  through 


472  A    MANUAL    OF    PHYSICS. 

which  a  current  i  flows  is  -idN,  where  <#N  is  the  increase  of  the 
number  of  the  lines  of  force  which  pass  through  the  circuit  in  the 
positive  direction.  Conversely,  the  work  which  is  done  upon  the 
circuit  by  the  electromagnetic  forces  during  a  process  in  which  the 
number  of  lines  of  force  which  pass  through  the  circuit  increases  by 
the  amount  dN  is  tdN.  If  the  conditions  are  such  that  this  work  can 
be  transformed  into  electric  energy  in  the  circuit,  a  reverse  electro- 
motive force  must  be  produced  which  opposes  the  passage  of  the 
current  i.  Now  this  electric  energy  is  developed  to  the  amount  £E 
per  unit  of  time,  where  E  is  the  reverse  electromotive  force  (§  342). 
Therefore,  since  dN/dt  is  the  increase  of  N  per  unit  of  time,  we  get 


that  is,  a  reverse  electromotive  force  acts  around  the  circuit  which 
at  any  instant  is  measured  by  the  rate  of  increase  of  the  number  of 
lines  of  force  which  pass  through  the  circuit.  [It  must  be  re- 
membered that  a  reverse  electromotive  force  is  one  which  tends  to 
produce  a  current  in  the  circuit  in  a  direction  which  is  related  to 
the  direction  of  the  lines  of  force  according  to  the  law  of  left-handed 
screwing  motion.] 

The  phenomenon,  whose  existence  we  have  here  assumed,  was 
discovered  experimentally  by  Faraday.  He  found  that  if,  from 
whatever  cause,  the  number  of  lines  of  force  passing  through  a  circuit 
is  increased,  a  reverse  electromotive  force  will  act  round  the  circuit, 
and  will  produce  a  reverse  current  ;  and  that,  if  the  number  of  lines  be 
decreased,  a  direct  force  will  act  and  will  produce  a  direct  current. 
The  currents  so  produced  are  called  induced  currents.  They  only 
last  so  long  as  there  is  a  variation  of  the  number  of  lines  of  force 
in  progress.  In  particular,  they  may  be  produced  by  the  electro- 
magnetic action  which  is  due  to  varying  currents  in  other  fixed 
circuits,  or  to  the  motion  of  other  circuits  which  carry  steady 
currents  ;  or  they  may  be  due  to  the  action  of  moving  magnets. 

For  example,  if  a  current  be  started  in  one  direction  in  a  linear 
conductor,  a  transient  current  will  flow  in  the  opposite  direction  in 
a  parallel  conductor  ;  and,  if  the  direct  current  in  the  former  be 
stopped,  a  transient  direct  current  will  flow  in  the  latter.  Pheno- 
mena such  as  these  are  caUed  phenomena  of  mutual  induction. 

But  it  is  important  to  observe  that  the  number  of  lines  of  force 
which  pass  through  a  closed  circuit  depends  upon  the  current  which 
is  flowing  through  that  circuit  as  well  as  upon  external  currents. 
These  lines  of  force  pass  in  the  positive  direction  through  the  cir- 
cuit, and  so  any  increase  in  their  number,  due  to  an  increase  in  the 


ELECTROMAGNETISM,    ETC.  473 

strength  of  the  current,  causes  a  reverse  electromotive  force  in  the 
circuit  which  prevents  the  direct  current  from  instantly  attaining  its 
full  strength  under  the  action  of  a  suddenly  introduced  electro- 
motive force,  or  from  instantly  falling  to  zero  when  the  force  is 
removed.  This  phenomenon  is  called  self-induction,  and  was 
investigated  experimentally  by  Faraday.  It  can  only  be  pre- 
vented by  arranging  the  circuit  in  such  a  way  that  its  total  area  is 
zero.  For  example,  if  a  plane  circuit  be  crossed  upon  itself  in  a 
figure-of-eight  shape,  so  that  the  areas  of  the  two  loops  are  equal, 
no  self-induction  will  occur,  for  the  lines  of  force  which  pass 
through  each  loop  are  equal  in  number,  but  are  oppositely  directed. 
The  number  of  lines  of  force  which  traverse  one  circuit  because 
of  the  electromagnetic  action  of  another  circuit  through  which  a 
current  j  is  flowing  is 

N=/M,  ...........  (2) 

where  M  is  a  quantity  which  depends  upon  the  form  and  mutual 
position  of  the  two  circuits  (§  362),  and  is  called  the  co-efficient  of 
mutual  induction  of  the  two  circuits. 

Let  a  constant  electromotive  force  J  act  in  the  circuit  through 
which  the  current  j  is  flowing,  and  let  an  electromotive  force  I  act 
in  another  circuit,  through  which  a  current  i  flows,  and  whose  co- 
efficient of  mutual  induction  with  regard  to  the  former  circuit  is  M. 
Also  let  the  resistance  of  the  former  circuit  be.  S,  while  that  of  the 
latter  is  K  ;  and  let  the  co-efficient  of  self-induction  of  the  former  be 
Q  ;  while  that  of  the  latter  is  P.  We  get 


If  the  two  circuits  are  fixed,  these  equations  become  respectively 


(6). 

wi/  14/lf 

The  first  term  on  the  right  hand  side  of  these  equations  represents 
the  reverse  electromotive  force  due  to  mutual  induction  ;  the  second 
represents  the  reverse  force  due  to  self-induction ;  and  the  third 
represents  (§  336)  the  part  of  the  electromotive  force  which  main- 
tains the  current  against  the  resistance  of  the  circuit. 


474  A    MANUAL    OF    PHYSICS. 

If  the  circuit  in  which  J  acts  be  entirely  removed,  the  equation 
which  applies  to  the  other  circuit  becomes 


Let  us  suppose  that  the  force  I  is  maintained  until  the  current 
attains  a  steady  value  iu,  after  which  I  is  withdrawn.  We  then  have 

P~+Ki  =  o, 

which  gives  (§38)  _  E 

i  =  iue 

This  shows  that  the  intensity  of  the  current  diminishes  in  geo- 
metrical progression  as  the  time  increases  in  arithmetical  progres- 
sion, and  that  it  (theoretically)  takes  an  infinite  time  to  reach  zero 
intensity.  Practically,  the  condition  of  zero  intensity  is  in  most 
cases  attained  in  a  small  fraction  of  a  second.  Similarly,  we  find 
that  the  equation 


represents  the  relation  between  the  current  i,  at  a  time  t  after  the 
electromotive  force  I  begins  to  act,  and  the  steady  current  iu. 

The  introduction  of  iron  cores  into  the  circuits  greatly  increases 
the  self  and  mutual  induction,  because  of  the  great  permeability  of 
iron.  This  is  the  essential  principle  of  the  Kuhmkorff  coil,  which 
consists  of  a  coil  of  stout,  insulated  copper  wire  wound  round  an  iron 
core,  and  surrounded  by  another  coil  of  very  fine,  well-insulated  copper 
wire.  The  inner  coil  is  called  the  primary  coil,  the  outer  is  called  the 
secondary  coil.  The  former  has  very  small  resistance,  while  the  latter 
has  very  high  resistance.  The  core  is  composed  of  a  number  of  fine 
iron  wires  for  the  purpose  of  preventing  the  induction  of  currents 
within  it,  for  these  currents  act  so  as  to  oppose  the  direct  induction. 
By  this  means  feeble  electromotive  forces  in  the  primary  circuit  may 
give  rise  to  very  high  electromotive  forces  in  the  secondary  circuit. 

In  the  modern  '  dynamo  '  coils  of  wire  with  iron  cores  are  caused 
to  move  rapidly  between  the  poles  of  a  powerful  electromagnet. 
Induced  currents  are  thus  produced  in  the  coils,  and  may  be  used 
for  purposes  of  electric  lighting,  etc.  We  cannot  here  enter  into  a 
discussion  of  the  various  ingenious  details  of  construction  which  are 
adopted  in  these  machines  in  order  to  secure  high  efficiency,  or  to 


ELECTROMAGNETISM,    ETC.  475 

adapt  them  for  the  performance  of  different  duties.  The  subject 
has  now  a  complete  literature  of  its  own. 

Arago  found  that  a  magnet,  which  is  pivoted  above  a  horizontal 
copper  disc,  will  be  set  into  rotation  if  the  disc  be  rotated.  Faraday 
explained  this  by  the  electromagnetic  action  of  the  currents  which 
are  induced  in  the  disc.  If  radial  slits  be  cut  in  the  disc,  the  action 
will  greatly  cease  ;  for  the  induction  of  currents  is  prevented  except 
on  a  small  scale. 

Currents  are  induced  in  the  body  of  a  magnet  itself  whenever  its 
state  of  magnetisation  varies.  Thus,  since  an  electric  current  flow- 
ing along  a  rod  is  surrounded  by  closed  lines  of  magnetic  force,  con- 
versely, any  change  in  the  circular  magnetisation  of  a  rod  will  cause 
the  flow  of  a  transient  current  along  the  rod.  This  may  readily  be 
made  manifest  by  connecting  the  ends  of  a  twisted  iron  rod  to 
the  terminals  of  a  galvanometer  (§  369),  and  suddenly  magnetising 
the  rod  longitudinally.  Since  the  rod  is  twisted,  longitudinal 
magnetisation  cannot  occur  without  circular  magnetisation  (§  357), 
and  so  a  transient  longitudinal  current  occurs,  and  is  made  manifest 
by  the  galvanometer.  The  same  effect  is  produced  if  a  longitudin- 
ally magnetised  rod  be  suddenly  twisted. 

368.  Electrokinetic  Energy.  —  Consider  again  the  two  electric 
circuits  dealt  with  in  last  section.  Let  them  move  so  that  M  in- 
creases by  the  amount  dM,  and  let  the  motion  be  so  slow  that  the 
currents  i  and  j  are  sensibly  constant.  The  rate  at  which  heat  is 
developed  per  unit  of  time  in  the  two  circuits  is 


and  the  rate  at  which  energy  is  supplied  per  unit  of  time  in  main- 
tainin   the  electromotive  forces  I  and  J  constant  is 


Hence  E-H=i(I-Ki)+y(J-S/)  ; 

which  becomes  E  -  H=2^'  —  , 

dt 

since,  in  (3)  and  (4)  above,  if  i,  j,  P,  and  Q  are  constant,  we  get 


,  T 

and  3- 

But  by  (1)  and  (2)  we  see  that 


..dM 


476  A   MANUAL    OF   PHYSICS. 

represents  the  rate  at  which  work  is  done  in  the  circuit  by  electro- 
dynamic  action,  and  thus  the  equation  shows  that,  under  the  given 
conditions,  an  amount  of  energy  must  be  drawn  from  the  source  in 
a  given  time  which  exceeds  that  developed  in  the  circuits  in  the 
form  of  heat  by  twice  the  amount  of  work  which  is  simultaneously 
performed  by  the  electromagnetic  forces.  This  excess  is  called  the 
electr akinetic  energy  of  the  system.  In  Maxwell's  theory  it  is 
supposed  to  reside  in  the  medium  which  surrounds  the  circuits. 
It  is  transformed  into  heat,  etc.,  whenever  the  circuits  are  broken  ; 
for  the  rupture  of  the  circuits  is,  under  these  conditions,  attended 
by  a  spark  of  more  than  usual  intensity. 

A  similar  result  can  be"  deducted  from  (5)  and  (6)  when  the 
circuits  are  fixed  and  i  andj  vary. 

369.  The  Galvanometer.  The  Ballistic  Method. — A  galvano- 
meter is  an  instrument  by  means  of  which  the  intensity  of  an 
electric  current  is  measured  through  the  magnetic  effect  which  the 
current  produces.  It  (as  already  stated)  usually  consists  of  coils  of 
wire  in  the  interior  of  which  a  small  magnet  is  freely  suspended.  In 
its  normal  position  the  magnet  has,  under  the  action  of  an  external 
force,  its  length  perpendicular  to  the  axis  of  the  coil ;  and  the  in- 
tensity of  a  current  is  proportional  to  the  tangent  of  the  angle 
through  which  the  magnet  is  deflected  from  its  normal  position 
when  a  current  passes  through  the  coil,  for  the  current  produces  in 
the  interior  of  the  coil  a  practically  uniform  magnetic  field,  whose 
intensity  is  proportional  to  the  current -strength  and  whose  direction 
is  parallel  to  the  axis  (§  365),  so  that  an  equation  of  the  form  (1), 
§  358,  applies. 

In  other  forms  of  the  instrument  the  coil  is  freely  suspended  in  a 
constant  magnetic  field ;  and  the  name  Electro  dynamometer  is 
given  to  an  instrument  in  which  both  of  the  magnetic  fields  are 
produced  by  the  current,  which  flows  simultaneously  through  two 
coils,  one  of  which  is  fixed,  while  the  other  is  freely  suspended  in 
its  interior  or  swings  freely  around  it.  The  indications  of  the 
latter  instrument  are  proportional  to  the  square  of  the  current - 
strength. 

In  the  Ballistic  Galvanometer  the  suspended  portion  has  great 
moment  of  inertia,  and,  consequently,  has  a  long  period  of  vibration 
(§  131).  When  a  transient  current,  whose  duration  is  very  small 
in  comparison  with  the  periodic  time,  passes  through  this  instru- 
ment, the  total  quantity  of  electricity  which  passes  is  proportional 
to  the  sine  of  half  the  angle  of  deflection.  Hence  the  instrument 
may  be  applied  to  the  investigation  of  magnetic  properties.  For 
example,  if  a  coil  of  wire  connected  with  the  galvanometer  be 


ELECTROMAGNETISM,    ETC.  477 

wound  on  an  iron  bar  the  intensity  of  magnetisation  of  which  is 
varied  from  time  to  time,  the  transient  current  which  follows  each 
variation  of  intensity  produces  a  deflection  in  terms  of  which  the 
total  quantity  of  electricity  which  passes  can  be  calculated.  Now, 
if  dN  be  the  change  of  induction  through  the  coil,  we  get  by  (1), 
§367, 


where  E  is  the  electromotive  force,  which  is  equal  to  Bi,  if  i  is  the 
intensity  of  the  current  and  B  is  the  resistance  of  the  circuit.  This 
gives,  as  the  total  change  of  induction,  the  quantity 

Hfidt, 

if  we  assume  that  K  is  constant.  But  this  is  equal  to  B#,  where  q 
is  the  total  quantity  of  electricity  which  has  passed.  Its  value  may, 
therefore,  be  determined  experimentally  by  means  of  the  indications 
of  the  galvanometer.  (It  must  be  remembered  that  if  the  coil 
consist,  for  example,  of  n  turns  wound  closely  on  the  bar,  the  actual 
induction  —  supposed  to  be  uniform  —  in  the  bar  is  ~Rqjn.)  Hence 
(§§  351,  352)  we  can  determine  the  permeability  and  susceptibility 
of  the  substance  of  which  the  bar  is  composed. 

370.  Electric  and  Magnetic  Units.  —  The  magnitude  of  electric 
and  magnetic,  as  of  all  other  quantities,  depends  upon  the  particular 
units  in  terms  of  which  they  are  measured.  All  such  quantities 
may  be  expressed  in  terms  of  the  units  of  mass,  length,  and  time  ; 
but  the  dimensions  of  a  quantity  in  terms  of  these  units  depends 
upon  the  particular  definition  of  some  electric  or  magnetic  quantity 
which  we  adopt. 

Two  systems  of  measurement  are  in  use  —  the  Electrostatic  and 
the  Electromagnetic.  In  the  electrostatic  system  we  start  from  the 
definition  that  two  similar  unit  quantities  of  electricity,  condensed 
at  points  which  are  at  unit  distance  apart,  repel  each  other  (in  air) 
with  unit  force  (§  312). 

The  dimensions  of  force  are  (§  64)  (MLT~2),  and,  therefore,  the 
dimensions  of  electric  quantity  are 


Surface  density  of  electricity  is  quantity  per  unit  surface.     Its 
dimensions  are,  therefore,  on  this  system, 


478  A   MANUAL   OF   PHYSICS. 

Electric  potential  and  electric  force  have  dimensions  (§  313) 

(tO^L-^CMWoT1) 
and 


respectively. 
The  dimensions  of  electrostatic  capacity  are  (§  314) 


those  of  current -strength  are  (§  335) 


and  those  of  resistance  are  (§  336)  directly  proportional  to  those  of 
potential  and  inversely  proportional  to  those  of  current-strength  ; 
they  are,  therefore, 

(K)=(L-1T). 

On  the  electromagnetic  system  the  definition  of  unit  quantity  of 
magnetism  is  precisely  analogous  to  the  definition  of  unit  quantity 
of  electricity  in  the  electrostatic  system.  Hence  the  dimensions  of 
quantity  of  magnetism,  surface  density  of  magnetism,  magnetic 
potential,  and  magnetic  force,  are,  on  this  system,  identical  with 
the  dimensions  of  the  corresponding  electric  quantities  on  the 
electromagnetic  system. 

In  addition,  the  dimensions  of  magnetic  moment  and  intensity 
of  magnetisation  on  the  latter  system  are  (§§  349,  351) 


and 


respectively.     The  latter  expression,  of  course,  is  identical  with  the 
expression  for  the  dimensions  of  surface  density. 

On  the  electromagnetic  system,  unit  current  is  the  current  which, 
flowing  in  a  circular  circuit  of  unit  radius,  exerts  unit  force  per  unit 
length  of  its  circumference  upon  a  unit  magnetic  pole  placed  at  its 
centre  (§  365).  Hence  the  dimensions  of  current  -strength  are 


The  quantity  of  electricity  which  is  conveyed  through  a  conductor 
is  directly  proportional  to  the  strength  of  the  current  and  to  the 
time  during  which  it  has  been  flowing.  Therefore,  its  dimensions  are 


ELECTROMAGNETISM,    ETC.  479 

The  dimensions  of  electric  potential  when  multiplied  by  quantity 
of  electricity  are  (§  320)  identical  with  those  of  energy,  and  are, 
therefore, 


By  such  considerations  we  may  readily  determine  the  dimensions 
of  any  electrical  or  magnetic  quantity  on  either  system  of  reckoning. 
Some  of  the  results  are  tabulated  below,  the  dimensions  on  the 
electrostatic  system  being  given  in  the  second  column,  while  those 
on  the  electromagnetic  system  are  given  in  the  third. 

Electrical  Quantities. 

Quantity  of  Electricity  .......  .'. 

Surface  density  of  Electricity 
Electric  Potential  ............... 

Electric  Force    .................. 

Electrostatic  Capacity  .........  (L)  (L      T") 

Current  Strength  ...............  (M  WT~  3  )         (M^L^T"  * 

Kesistance  ........................  (L~lrr)  (LT"1) 

Specific  Inductive  Capacity...  (M°L°T°)  (IT2T2) 

Magnetic  Quantities. 

Quantity  of  Magnetism  ......  (M^lJ") 

Surface  density  of  Magnetism  (M^TT^ 

Magnetic  Potential   ............  (M^lA? 

Magnetic  Force  ..................  (M^T 

Magnetic  Moment    ............ 

Intensity  of  Magnetisation  ... 

Magnetic  Permeability     ......  (L~2T2)  (M°L0T°) 

Magnetic  Susceptibility   ......  (L~2T2)  (M0L°T0) 

It  is  specially  worthy  of  notice  that  the  dimensions  of  any 
quantity  on  one  or  other  of  these  systems  always  differ  from 
its  dimensions  on  the  other  by  the  dimensions  of  a  speed  or  of  a 
speed  squared. 


480  A   MANUAL    OF    PHYSICS. 

The  force  between  two  quantities,  q  and  g",  of  electricity  at  a 
distance  r  apart  in  a  medium  other  than  air,  is  qq'Kr2,  where  K  is 
the  specific  inductive  capacity.  Hence,  if  we  choose  not  to  define 
the  electrostatic  dimensions  of  K  as  zero,  but  leave  them  undeter- 
mined, the  electrostatic  dimensions  of  quantity  of  electricity  become 


Similarly,  if  we  leave  the  dimensions  of  magnetic  permeability  (/*) 
undetermined,  we  find  that  the  electromagnetic  dimensions  of 
quantity  of  electricity  are 


and  the  corresponding  alterations  in  the  dimensions  of  other  quanti- 
ties can  easily  be  found.  One  advantage  of  this  method  (due  to 
Eiicker)  is,  as  Fitzgerald  pointed  out,  that  we  can  make  the  dimen- 
sions of  any  one  quantity  on  both  systems  identical  by  assuming 
that  the  dimensions  of  K  and  /*  are  (TL-1). 

It  is  convenient,  in  scientific  measurements,  to  adopt  the  cen- 
timetre-gramme-second (c.g.s.)  system  of  units  ;  but,  in  practice, 
these  units  are  often  inconveniently  large  or  inconveniently  small. 
In  the  following  table  the  name  of  the  practical  unit  of  various 
quantities  is  given  in  the  second  column  ;  and  the  factors  which  are 
required  to  reduce  the  numerics,  as  expressed  in  terms  of  the  prac- 
tical units,  to  their  equivalents  on  the  c.g.s.  system,  are  given  in  the 
third  column. 

Quantity  of  Electricity  .........     Coulomb       .........  10—  l 

Electromotive  Force  ............     Volt  .........  10s 

Electrostatic  Capacity  .........     Farad  .........  10~y 

Microfarad  .........  10~15 

Current  Strength   ...............     Ampere         .........  10"1 

Resistance  ........................     Ohm  .........  10» 

An  electromotive  force  of  one  volt  maintains  a  current,  whose 
strength  is  one  ampere,  through  a  resistance  of  one  ohm. 


CHAPTER  XXXII. 

ELECTROMAGNETIC    THEORY    OF   LIGHT. 

371.  Magnetic  Rotation  of  the  Plane  of  Polarisation  of  Light. — 
Faraday  made  many  attempts  to  detect  some  action  upon  polarised 
light  when  it  was  made  to  pass  through  a  dielectric  which  was  sub- 
jected to  electric  stress.  He  also  sought  for  evidence  of  such  action 
when  polarised  light  passed  through  an  electrolyte  conveying  a 
current,  but  in  no  case  could  he  observe  any  effect.  On  the  other 
hand,  he  found  a  marked  effect  when  polarised  light  was  passed 
through  a  diamagnetic  medium  placed  in  a  field  of  magnetic  force. 

When  the  direction  of  the  ray  coincides  with  the  positive  direction 
of  the  lines  of  force,  the  plane  of  polarisation  is  rotated  through  an 
angle  which  is  proportional  to  the  intensity  of  the  magnetic  field, 
and  to  the  length  of  the  path  of  the  ray  within  the  medium.  If  the 
direction  of  the  ray  does  not  coincide  with  the  direction  of  the  field, 
the  rotation  is  proportional  to  the  intensity  of  the  resolved  part  of 
the  force  taken  in  the  direction  of  the  ray.  The  amount  of  the 
rotation  per  unit  of  length,  in  a  field  of  unit  intensity,  depends  upon 
the  nature  of  the  medium.  The  absolute  direction  of  rotation  is 
unaltered  by  a  reversal  of  the  ray,  provided  that  the  direction  of 
the  field  is  unaltered. 

In  diamagnetic  media  the  direction  of  the  rotation  is,  in  general, 
connected  with  that  of  the  field  according  to  the  law  of  right-handed 
screwing  motion ;  in  paramagnetic  media,  the  reverse  is  generally 
true. 

The  fact  of  the  non-reversal  of  the  rotation  with  reference  to  the 
direction  of  the  field  points  to  a  fundamental  distinction  between 
the  mechanical  method  by  which  this  magnetic  rotation  is  produced 
and  that  obtained  in  cases  of  rotation  by  quartz  or  solutions  such  as 
sugar  (§  251).  In  the  latter  cases,  reversal  of  the  ray  is  not  accom- 
panied by  a  reversal  of  the  rotation  with  reference  to  it ;  and  thus 
the  total  rotation  during  the  double  passage  through  the  medium 

31 


482  A    MANUAL    OF    PHYSICS. 

'is  zero.  The  total  magnetic  rotation  is  doubled  by  the  double 
passage. 

In  the  case  of  vapours  and  gases,  the  rotation  is  very  small  in 
comparison  with  the  rotation  which  is  produced,  under  similar  con- 
ditions, by  solids  and  liquids.  Even  in  vapours  of  liquids  wThich 
have  considerable  rotatory  power,  the  effect  is  very  small. 

Verdet  has  shown  that  the  rotation  is  approximately  inversely 
proportional  to  the  square  of  the  wave-length — the  deviation  being 
in  defect  as  the  wave-length  increases,  and  being  most  marked  in 
substances  of  great  dispersive  power. 

372.  Hypothesis  of  Molecular  Vortices. — We  have  seen  that  a 
plane  polarised  ray  may  be  compounded  of  two  uniform  equi- 
periodic  circular  motions  of  equal  amplitude,  and  that  rotation  of 
the  plane  of  polarisation  will  take  place  if  one  of  these  component 
motions  is  accelerated  relatively  to  the  other  (£  251).  The  explana- 
tion of  the  magnetic  effect  seems  to  lie  in  this  direction. 

Sir  W.  Thomson  has  remarked  on  this  subject  '  That  the  magnetic 
influence  on  light  discovered  by  Faraday  depends  on  the  direction 
of  motion  of  moving  particles.  For  instance,  in  a  medium  possess- 
ing it,  particles  in  a  straight  line  parallel  to  the  lines  of  force, 
displaced  to  a  helix  round  this  line  as  axis,  and  then  projected 
tangentially  with  such  velocities  as  to  describe  circles,  will  have 
different  motions  according  as  their  motions  are  round  in  one  direc- 
tion (the  same  as  the  nominal  direction  of  the  galvanic  current  in 
the  magnetising  coil),  or  in  the  contrary  direction.  But  the  elastic 
reaction  of  the  medium  must  be  the  same  for  the  same  displace- 
ments, whatever  be  the  velocities  and  directions  of  the  particles ; 
that  is  to  say,  the  forces  which  are  balanced  by  centrifugal  force  of 
the  circular  motions  are  equal,  while  the  luminiferous  motions  are 
unequal.  The  absolute  circular  motions  being  therefore  either  equal 
or  such  as  to  transmit  equal  centrifugal  forces  to  the  particles 
initially  considered,  it  follows  that  the  luminiferous  motions  are 
only  components  of  the  whole  motion,  and  that  a  less  luminiferous 
component  in  one  direction,  compounded  with  a  motion  existing  in 
the  medium  when  transmitting  no  light,  gives  an  equal  resultant  to 
that  of  a  greater  luminiferous  motion  in  the  contrary  direction  com- 
pounded with  the  same  non-luminous  motion.  I  think  it  not  only 
impossible  to  conceive  any  other  than  this  dynamical  explanation  of 
the  fact  that  circularly -polarised  light  transmitted  through  mag- 
netised glass  parallel  to  the  lines  of  magnetising  force,  with  the 
same  quality,  right-handed  always,  or  left-handed  always,  is  pro- 
pagated at  different  rates  according  as  its  course  is  in  the  direction, 
or  is  contrary  to  the  direction,  in  which  a  north  magnetic  pole  is 


ELECTROMAGNETIC    THEORY    OF   LIGHT.  483 

drawn  ;  but  I  believe  it  can  be  demonstrated  that  no  other  ex- 
planation of  that  fact  is  possible.  Hence  it  appears  that  Faraday's 
optical  discovery  affords  a  demonstration  of  the  reality  of  Ampere's 
explanation  of  the  ultimate  nature  of  magnetism  ;  and  gives  a  defi- 
nition of  magnetisation  in  the  dynamical  theory  of  heat.  The 
introduction  of  the  principle  of  moments  of  momenta  ("the  conser- 
vation of  areas  ")  into  the  mechanical  treatment  of  Mr.  Eankine's 
hypothesis  of  "  molecular  vortices  "  (see  ^  254),  appears  to  indicate  a 
line  perpendicular  to  the  plane  of  resultant  rotatory  momentum 
("  the  invariable  plane  ")  of  the  thermal  motions  as  the  magnetic 
axis  of  a  magnetised  body,  and  suggests  the  resultant  moment  of 
momenta  of  these  motions  as  the  definite  measure  of  the  "  magnetic 
moment."  The  explanation  of  all  phenomena  of  electromagnetic 
attraction  and  repulsion,  and  of  electromagnetic  induction,  is  to  be 
looked  for  simply  in  the  inertia  and  pressure  of  the  matter  of  which 
the  motions  constitute  heat.  Whether  this  matter  is  or  is  not  elec- 
tricity, whether  it  is  a  continuous  fluid  interpermeating  the  spaces 
between  molecular  nuclei,  or  is  itself  molecularly  grouped;  or 
whether  all  matter  is  continuous,  and  molecular  heterogeneousness 
consists  in  finite  vortical  or  other  relative  motions  of  contiguous 
parts  of  a  body,  it  is  impossible  to  decide,  and  perhaps  in  vain  to 
speculate,  in  the  present  state  of  science.' 

The  idea  contained  in  these  remarks  has  been  developed  by  Max- 
well into  a  complete  theory  of  molecular  vortices.  He  points  out 
that,  from  the  fact  that  the  wave-length,  X,  and  the  periodic  time,  r, 
increase  and  decrease  together,  it  follows  that  if  for  a  given  nume- 
rical value  of  the  angular  velocity,  n  (  =  27r/r),  the  value  of  the  speed 
of  propagation,  X/r,  is  greater  when  n  is  positive  than  when  it  is 
negative,  for  a  given  value  of  X  the  positive  value  of  n  will  be  greater 
than  the  negative  value.  This  is  so  since  the  former  condition  im- 
plies that  X  is  greater  when  n  is  positive  than  when  it  is  negative, 
and  a  diminution  of  X  implies  a  diminution  of  r  and  therefore  an 
increase  of  n.  Since  the  ray  does  not  diminish  in  intensity  as  it 
passes  through  the  medium,  the  amplitude,  r,  must  remain  constant 
(§  179)  ;  and  the  principle  of  conservation  of  energy  shows  that,  for 
equilibrium,  we  must  have  the  condition 


where  T  and  V  are  respectively  the  kinetic  and  potential  energies. 
But  the  expression  for  T  contains  one  term  involving  ri2  ;  and  it  may 
contain  terms  involving  the  products  of  n  into  other  velocities,  and 

31—2 


484  A    MANUAL    OF    PHYSICS- 

terms  independent  of  n.     V,  on  the  other  hand,  is  independent  of  n. 
Hence  the  above  equation  is  of  the  form 


where  A,  B,  and  C,  are  functions  of  the  co-ordinates.  Now  ex- 
periment shows  that  n  has  two  real  values,  one  positive,  the  other 
negative  and  smaller.  C  must  therefore  be  finite,  and  both  it  and 
B  must  be  negative  if  A  is  positive;  for  —  B/A  and  C/A  are  re- 
spectively the  sum  and  the  product  of  the  roots  of  the  equation,  and 
the  sum  is  positive  while  the  product  is  negative.  B  also  cannot 
vanish,  since  the  roots  are  distinct.  The  term  in  n  must  there- 
fore involve  another  velocity  besides  n,  and  that  velocity  must 
be  an  angular  velocity  about  the  same  axis,  for  Bn  is  a  scalar 
quantity. 

Maxwell  then  concludes  '  That  in  the  medium,  when  under  the 
action  of  magnetic  force,  some  rotatory  motion  is  going  on,  the  axis 
of  rotation  being  in  the  direction  of  the  magnetic  forces  ;  and  that 
the  rate  of  propagation  of  circularly  polarised  light,  when  the  direc- 
tion of  its  vibratory  rotation  and  the  direction  of  the  magnetic 
rotation  of  the  medium  are  the  same,  is  different  from  the  rate  of 
propagation  when  these  directions  are  opposite. 

'  The  only  resemblance  which  we  can  trace  between  a  medium 
through  which  circularly-polarised  light  is  propagated,  and  a  medium 
through  which  lines  of  magnetic  force  pass,  is  that  in  both  there  is 
a  motion  of  rotation  about  an  axis.  But  here  the  resemblance  stops, 
for  the  rotation  in  the  optical  phenomenon  is  that  of  the  vector 
which  represents  the-  disturbance.  This  vector  is  always  perpen- 
dicular to  the  direction  of  the  ray,  and  rotates  about  it  a  known 
number  of  times  in  a  second.  In  the  magnetic  phenomenon,  that 
which  rotates  has  no  properties  by  which  its  sides  can  be  dis- 
tinguished, so  that  we  cannot  determine  how  many  times  it  rotates 
in  a  second. 

'  There  is  nothingj  therefore,  in  the  magnetic  phenomenon  which 
corresponds  to  the  wave-length  and  the  wave-propagation  in  the 
optical  phenomenon.  A  medium  in  which  a  constant  magnetic 
force  is  acting  is  not,  in  consequence  of  that  force,  filled  with  waves 
travelling  in  one  direction,  as  when  light  is  propagated  through  it. 
The  only  resemblance  between  the  optical  and  the  magnetic  pheno- 
menon is,  that  at  each  point  of  the  medium  something  exists  of  the 
nature  of  an  angular  velocity  about  an  axis  in  the  direction  of  the 
magnetic  force.' 

.  *  This  angular  velocity  cannot  be  that  of  any  portion  of  the  medium 
of  sensible  dimensions  rotating  as  a  whole.     We  must  therefore  con- 


ELECTROMAGNETIC   THEORY   OF   LIGHT.  485 

ceive  the  rotation  to  be  that  of  very  small  portions  of  the  medium, 
each  rotating  on  its  own  axis.  This  is  the  hypothesis  of  molecular 
vortices. 

'  The  motion  of  these  vortices,  though  ...  it  does  not  sensibly 
affect  the  visible  motions  of  large  bodies,  may  be  such  as  to  affect 
that  vibratory  motion  on  which  the  propagation  of  light,  according 
to  the  undulatory  theory,  depends.  The  displacements  of  the 
medium,  during  the  propagation  of  light,  will  produce  a  disturbance 
of  the  vortices,  and  the  vortices  when  so  disturbed  may  react  on  the 
medium  so  as  to  affect  the  mode  of  propagation  of  the  ray.' 

From  this  hypothesis  Maxwell  has  deduced  an  expression  for  the 
magnitude  of  the  rotation  under  given  conditions  which  accords  very 
well  with  the  results  of  observation. 

373.  Hall's  Effect. — Hall  found  by  experiment  that  a  thin  metallic 
conductor,  which  is  placed  in  a  magnetic  field  of  force  with  its 
plane  perpendicular  to  the  lines  of  force,  and  through  which  an 
electric  current  flows,  is  the  seat  of  an  electromotive  force  which 
acts  along  the  common  perpendicular  to  the  directions  of  the  current 
and  the  magnetic  field.      Kowland  has  proved  that,  if  a  similar 
electromotive  force  appears  when  the  electric  displacement  in  an 
insulating  medium  varies  in  a  field  of  magnetic  force,  rotation  of 
the  direction  of  the  displacement  will  follow  the  passage  of  a  wave 
through  the  medium  ;  and  Glazebrook  has  shown  that  the  electro- 
motive force  is  a  consequence  of  the  molecular  rotation  which  Max- 
well assumes. 

374.  Kerr's  Effects. — The  action  upon  polarised  light  in  a  medium 
subjected  to  electric  stress,  for  which  Faraday  sought  in  vain,  was 
discovered  by  Kerr.     He  placed  two  parallel  brass  plates  at  a  short 
distance  apart,  in  a  glass  cell  containing  carbon  bisulphide,  and 
connected  them  with  the  poles  of  an  electric  machine.     A  beam  of 
light,  polarised  at  an  angle  of  45°  to  the  direction  of  the  lines  of 
electric  force,  was  passed  between  the  plates,  and,  on  emergence, 
was  found  to  be  elliptically  polarised.     The  axes  were  respectively 
parallel  to  and  perpendicular  to  the  lines  of  force ;  and  the  differ- 
ence between  the  phases  of  the  two  components  was  proportional  to 
the  square  of  the  intensity  of  the  electric  field. 

Another  form  of  this  experiment  consists  in  passing  the  polarised 
beam  between  two  small  spheres  which  are  placed  near  to  each 
other  in  the  insulating  medium,  and  are  connected  to  the  poles  of 
an  electric  machine.  The  medium  is  then  found  to  have  become 
doubly  refractive. 

Dr.  Kerr  has  also  discovered  that  the  plane  of  polarisation  of  light 
is  rotated  in  the  act  of  reflection  from  the  (highly-polished)  pole  of 


486  A    MANUAL    OF    PHYSICS. 

an  electromagnet  —  the  direction  of  the  rotation  being  reversed  when 
the  magnetisation  of  the  pole  is  reversed. 

375.  Electromagnetic  Theory  of  Light.  —  The  phenomena  which 
are  described  in  the  immediately  preceding  sections  point  to  a  very 
close  connection  between  electricity,  magnetism,  and  light. 

We  have  already  seen  that  the  phenomena  of  light  are  best  ex- 
plained on  the  assumption  that  light  consists  in  undulations  pro- 
pagated through  a  medium.  On  the  other  hand,  the  theories  of 
electrical  and  magnetic  action  were  originally  expressed  in  terms  of 
direct  action  at  a  distance  ;  and  (although  Faraday  conducted  all 
his  reasoning  on  the  assumption  of  action  through  a  medium)  it  was 
not  until  Maxwell  translated  Faraday's  ideas  into  mathematical 
language  that  it  was  recognised  that  electrical  and  magnetic  pheno- 
mena could  be  readily  explained  as  the  results  of  the  propagation  of 
action  through  a  medium. 

In  determining  the  conditions  of  the  propagation  of  an  electro- 
magnetic disturbance  through  the  medium  whose  existence  he 
postulated,  Maxwell  arrived  at  the  conclusion  that  the  propagation 
takes  place  in  accordance  with  the  laws  of  the  transference  of 
motion  through  an  elastic  solid,  and  that  the  speed  of  propaga- 
tion is 

v= 


where  K  and  ^t  are  respectively  the  specific  inductive  capacity  and 
the  permeability  of  the  medium.  In  the  propagation  of  a  plane 
wave,  electric  displacement  takes  place  at  right  angles  to  the  direc- 
tion of  magnetic  induction,  and  both  are  in  the  plane  of  the  wave. 

A  reference  to  §  370  will  make  it  evident  that",  on  either  of  the 
electrostatic  or  the  electromagnetic  systems,  the  dimensions  of 
K  are  those  of  the  inverse  square  of  a  speed.  According  to  Max- 
well,  if  light  is  an  electromagnetic  phenomenon,  V  must  represent 
the  speed  of  light.  Now  the  speed  of  light,  v,  is  capable  of 
measurement  to  a  considerable  degree  of  accuracy,  while  the  value 
of  V  can  be  determined  directly  by  a  comparison  of  the  relative 
values  of  some  electrical  or  magnetic  quantity  on  the  two  systems 
of  measurement  (§  370)  ;  and  the  results  of  various  independent 
determinations  of  the  values  of  V  and  v  strongly  confirm  the 
supposition  of  their  numerical  identity  in  air. 

In  media  other  than  air  the  speed  of  light  is  inversely  propor- 
tional to  the  refractive  indices.  Hence  K  should  be  practically 
equal  to  the  square  of  the  refractive  index,  for  the  value  of  p  is 
nearly  unity  in  all  transparent  media.  Since  the  experimental 


ELECTROMAGNETIC    THEORY   OF   LIGHT.  487 

determination  of  K  occupies  a  time  which  is  practically  infinite 
in  comparison  with  the  period  of  any  luminous  vibration,  we  must, 
in  testing  this  point,  take  the  refractive  index  for  rays  of  infinite 
wave-length.  (This  might  be  given  by  the  value  of  the  constant  a 
in  Cauchy's  expression  for  the  refractive  index,  §  209).  Hopldnson 
has  found  that  while  the  relation  holds  in  the  case  of  hydrocarbons, 
it  does  not  obtain  in  glass  and  the  animal  and  vegetable  oils.  In 
the  latter  substances  the  refractive  index  is  less  than  \/H. 

In  the  luminiferous  medium,  two  forms  of  energy  exist — one 
kinetic,  the  other  potential.  Similarly,  in  the  electromagnetic 
medium,  energy  exists  in  a  kinetic  (electrokinetic,  §  368)  form  and 
in  a  potential  (electrostatic,  §  320)  form. 

The  theory  also  explains  double  refraction,  and  leads  to  Fresnel's 
construction  for  the  wave  surface.  And  it  shows  that  the  speed 
of  propagation  of  a  condensational-rarefactional  wave  would  be 
infinite ;  so  that  this  wave  does  not  exist — a  result  which  is  in 
harmony  with  optical  observations.  Also,  if  the  medium  be  not  a 
perfect  conductor,  the  electrical  energy  is  in  part  transformed  into 
energy  of  electric  currents,  and  so  finally  into  heat.  This  explains 
the  absorption  of  light. 

376.  Electromagnetic  Waves. — The  above  evidence  in  favour  of 
the  truth  of  Maxwell's  electromagnetic  theory  is  very  strong  in 
itself.  But  recent  investigations,  by  Hertz  and  others,  have  proved, 
beyond  the  possibility  of  doubt,  that  electromagnetic  action  is 
propagated  with  finite  speed  through  a  medium,  and  have  indicated 
that  its  speed  of  propagation  is  identical  with  that  of  light. 

If  the  initial  disturbance  is  periodic,  a  series  of  electromagnetic 
waves  are  propagated  outwards  from  the  source.  The  condition  of 
periodicity  can  be  obtained  by  means  of  the  disruptive  discharge, 
which,  under  suitable  conditions,  is  oscillatory  in  its  nature  (§  321) 
and  has  a  constant  period  of  oscillation  depending  on  the  electro- 
static capacity  and  the  coefficient,  of  self-induction  of  the  apparatus 
which  is  used  for  the  production  of  the  discharge. 

Let  us  suppose,  for  the  sake  of  definiteness,  that  the  discharge 
takes  place  between  the  poles  of  a  Holtz  machine.  The  alternating 
currents  which  characterise  the  discharge  will  induce  similar 
currents  in  neighbouring  conductors.  According  to  the  theory  of 
direct  action  at  a  distance,  these  induced  currents  will  appear  in 
exact  simultaneity  with  the  inducing  currents ;  according  to  the 
electromagnetic  theory,  they  will  appear  later  and  later  in  propor- 
tion as  the  conductors  in  which  they  are  induced  are  more  and 
more. remote  from  the  Holtz  machine. 

In  his   investigations   on  this  point,   Hertz   made  use   of    the 


488  A   MANUAL   OF   PHYSICS. 

principle  of  resonance  (cf.  §  173).  That  is,  he  used  a  secondary 
conducting  circuit  the  natural  period  of  electric  oscillation  in  which 
was  the  same  as  that  in  the  primary  circuit.  The  result  is  that  the 
magnitude  of  the  induced  oscillations  may  be  made  very  great,  for 
each  succeeding  induced  oscillation  is  timed  to  re-enforce  the  effects 
of  preceding  oscillations.  In  this  way  sufficient  electromotive  force 
may  be  developed  to  cause  the  electricity  to  spark  across  a  small 
air-gap  in  the  circuit. 

Such  a  circuit,  with  its  air-gap,  may  be  used  to  make  evident 
the  existence  of  inductive  effects  at  any  given  point  in  space,  pro- 
vided that  its  distance  from  the  source  is  not  too  great.  If  one 
source  alone  existed  in  space,  the  intensity  of  the  inductive  effect  at 
any  given  point,  and  therefore  the  intensity  of  the  spark  in  the 
secondary  circuit,  would  diminish  continuously  as  the  distance 
between  the  point  and  the  source  increased.  If  two  sources  existed, 
the  intensity  might  be  great  in  the  neighbourhood  of  each  and 
might  reach  a  minimum  at  some  intermediate  position :  or,  if  the 
effects  of  the  sources  were  opposite,  the  intensity  of  the  resultant 
effect  might  be  zero  at  some  point  between  the  two.  If  the  effects 
were  instantaneously  propagated,  only  one  such  minimum  could 
exist.  But,  if  the  effects  were  propagated  by  wave-motion  at  a 
finite  rate,  a  great  number  of  maxima  and  minima  might  appear, 
in  accordance  with  the  ordinary  laws  of  interference.  The  best 
results  will  be  obtained  when  the  two  sources  are  precisely  similar. 
Now,  if  a  conducting  sheet  be  placed  in  the  neighbourhood  of  a 
single  source,  the  currents  which  are  induced  in  it  will,  in  turn, 
give  rise  to  electromagnetic  effects  having  a  periodic  time  equal  to 
that  of  the  source.  This,  on  the  electromagnetic  theory,  constitutes 
reflection  of  the  electromagnetic  radiation,  and  interference  may  be 
expected  to  take  place  between  the  incident  and  the  reflected  waves 
— nodes  and  loops  occurring  alternately  at  equal  intervals  of  one 
half  of  the  length  of  a  wave  (§53).  In  performing  this  experi- 
ment Hertz  was  able  to  observe  the  existence  of  successive  maxima 
and  minima,  and  so  the  existence  of  electromagnetic  radiation  was 
proved. 

If  the  original  electrical  oscillations  take  place  along  a  straight 
rod,  the  oscillations  in  the  electromagnetic  medium  will  be  parallel 
to  the  axis  of  the  rod  ;  i.e.,  the  wave  is  plane  polarised.  And  the 
rod  is  surrounded  by  circular  lines  of  magnetic  induction,  the 
direction  of  the  induction  changing  with  each  alternation  of  electric 
displacement.  The  electric  displacement  and  the  magnetic  induc- 
tion therefore  take  place  in  the  front  of  the  wave,  and  are  at  right 
angles  to  each  other ;  and  these  two  effects  can  be  separated  from 


ELECTROMAGNETIC    THEORY    OF    LIGHT.  489 

each  other  by  using  a  suitable  resonating  circuit.  A  circular 
circuit  held  with  its  plane  passing  through  the  axis  of  the  rod  and 
its  spark-gap  at  right  angles  to  the  axis  will  respond  only  to  the 
magnetic  variations.  On  the  other  hand,  if  held  in  front  of  the 
axis  with  the  line  of  its  spark-gap  and  one  of  its  diameters  parallel 
to  the  axis,  it  will  respond  only  to  the  electrical  variations,  for  no 
lines  of  magnetic  induction  pass  through  it. 

With  apparatus  such  as  is  ordinarily  used  in  a  laboratory,  radia- 
tions having  a  wave-length  varying  from  a  few  inches  to  a  number 
of  miles  in  length,  can  be  readily  obtained.  These  long-period 
radiations  can  pass  freely  through  insulators,  such  as  pitch,  which 
are  opaque  to  luminous  radiations.  And,  by  using  large  prisms  of 
such  substances,  their  refractive  indices  can  be  found  by  the  usual 
methods. 

If  the  radiation  falls  upon  a  transparent  sheet  of  thickness  which 
is  small  in  comparison  with  the  wave  -  length,  no  reflection  is 
observed ;  for  the  acceleration  of  the  phase  by  half-a-period,  which 
takes  place  at  the  second  surface,  produces  total  interference.  This 
is  a  reproduction  of  the  phenomenon  of  the  central  black  spot  in 
Newton's  rings  (§  221). 

Keflection  can  be  obtained  at  the  surface  of  a  fhick  insulator  if 
the  direction  of  the  electric  displacements  is  perpendicular  to  the 
plane  of  reflection ;  but  none  is  found  at  the  polarising  angle  if  the 
line  of  displacement  lies  in  the  plane  of  reflection.  This  proves  that 
the  electric  displacement  takes  place  at  right  angles  to  the  plane  of 
polarisation,  and  settles  the  much  debated  question  of  the  direction 
of  the  luminous  vibrations  in  favour  of  Fresnel's  assumption 
(§  239). 

Effects  of  diffraction  can  also  be  observed  with  these  waves,  and 
are  in  strict  accordance  with  the  results  of  the  undulatory  theory. 

Maxwell  concluded  from  his  theory  that  a  body  which  absorbs 
light  should  be  repelled  towards  the  unilluminated  side.  The  effect 
is  too  small  for  observation  with  luminous  rays — even  with  con- 
centrated sunlight ;  but  a  disc  of  good  conducting  silver  is  repelled 
from  the-  pole  of  an  electromagnet  which  is  excited  by  a  powerful 
alternating  current. 


CHAPTER  XXXIII. 

THE    ETHER. 

377.  WHENEVER  mutual  action  is  observed  between  two  systems,  it 
is  possible  to  explain  the  accompanying  phenomena,  in  great  part  at 
least,  on  either  of  two  assumptions.  We  may  assume  that  the 
action  occurs  directly  at  a  distance,  or  we  may  assume  that  it  is 
propagated  by  means  of  a  material  medium.  But  when  we  can 
show  that  the  action  takes  time  to  travel  from  one  point  of  space  to 
another,  the  latter  assumption  only  is  tenable.  For  example,  we 
have  no  evidence  as  yet  (§  92)  that  gravitational  action  is  not  in- 
stantaneously propagated,  and  therefore  either  assumption  is  valid ; 
but  the  experiments  of  Hertz  have  shown  that  electrodynamic 
action  requires  a  finite  time  for  its  propagation  over  a  finite  space, 
and  therefore  we  must  assume  the  existence  of  a  medium  by  means 
of  which  that  action  is  transferred. 

Such  a  medium  is  termed  an  '  ether,'  and  we  may  therefore 
define  the  ether  as  a  substance,  other  than  ordinary  matter,  through 
which  action  is  propagated. 

At  one  time  many  ethers  were  supposed  to  exist — indeed,  a  new 
ether  was  postulated  for  the  explanation  of  almost  every  new  class 
of  phenomena  which  presented  itself.  An  ether  was  assumed  in 
order  to  explain  gravitation.  Newton  introduced  a  medium  to  ac- 
count for  the  production  of  the  '  fits  '  of  easy  reflection  and  transmis- 
sion of  luminous  corpuscles,  which  he  had  to  postulate  for  the  expla- 
nation of  the  colours  of  thin  plates,  etc.,  on  the  corpuscular  theory. 
The  aid  of  another  was  invoked  for  the  explanation  of  physiological 
phenomena,  and  so  on — the  attributes  with  which  each  was  endowed 
being  specially  chosen  in  order  to  make  it  suit  each  particular  case. 

Such  a  procedure  is  totally  unscientific,  and  it  is  now  the  aim  of 
scientists  to  ascribe  all  actions,  which  apparently  occur  directly  at  a 
distance,  to  the  intervention  of  a  single  medium.  Of  all  the  host  of 
mediaeval  ethers,  one  alone  remains — the  medium  whose  existence 
was  postulated  by  Huyghens  in  his  explanation  of  the  phenomena 


THE    ETHER.  491 

of  light ;  and  one  of  the  great  merits  of  Maxwell's  modern  electro- 
magnetic ether  is  that  it  explains  the  phenomena  of  light  as  well  as 
the  phenomena  of  electricity  and  magnetism. 

378.  It  is  more  easy  to  say  what  the  ether  is  not  than  to  say 
what  it  is.  It  is  not  a  gas,  like  air  ;  for  transverse  oscillations,  such 
as  those  which  take  place  in  the  propagation  of  light,  die  out  with 
extreme  rapidity  in  such  a  medium.  Neither,  for  a  like  reason,  is 
it  a  liquid  like  water.  So  far  as  this  effect  is  concerned,  it  might  be 
a  transparent  solid,  for  solids  can  transmit  transverse  oscillations. 
But  on  the  other  hand,  the  rate  at  which  transparent  solids  transmit 
such  vibrations  is  immensely  slower  than  the  rate  at  which  the 
ether  transmits  light.  Hence  the  ether  cannot  be  an  ordinary 
transparent  solid,  although  it  interpenetrates  such  solids,  and  is 
hampered  in  its  action  by  them — a  fact  shown  by  the  diminished 
speed  of  light  when  passing  through  them. 

Yet  the  ether,  although  it  is  not  ordinary  matter,  must  be 
material,  i.e.,  must  possess  inertia,  for  it  transmits  energy  at  a 
finite  rate.  And,  for  the  same  reason,  it  must  possess  rigidity  and 
must  be  elastic.  The  rate  of  vibration  of  the  parts  of  the  medium 
is  (cf.  $  168)  directly  proportional  to  the  square  root  of  the  rigidity, 
and  is  inversely  proportional  to  the  square  root  of  the  density. 
When  red  light  passes  through  the  ether,  the  rate  of  vibration  is 
about  400,000,000  times  per  second.  A  steel  tuning-fork  which 
emits  even  the  highest  audible  note  would  require  to  be  immensely 
more  rigid  than  it  is  if  it  were  to  vibrate  at  that  rate ;  if  its  rigidity 
remained  constant,  it  would  require  to  be  far  less  massive  than  it  is. 
It  would  appear  from  Thomson's  calculations,  based  on  a  very 
plausible  assumption,  that  the  ether  is  about  (10)9  times  less  rigid 
than  steel  is ;  but,  on  the  other  hand,  its  density  would  appear  to 
be  about  (10)19  times  less  than  that  of  steel. 

379.  It  seems,  therefore,  that  the  ether  acts  as  if  it  were  an 
elastic  solid.  And  yet  the  earth,  in  its  course  round  the  sun,  moves 
through  it  without  being  subjected  to  any  appreciable  resistance ; 
and  the  light  which  comes  to  us  from  a  distant  star  gives  no  evi- 
dence of  the  disturbing  effect  which  the  earth  might  be  supposed  to 
have  upon  the  ether  in  its  neighbourhood  as  it  moves  quickly 
through  it. 

Young  therefore  suggested  that  the  structure  of  the  earth  and 
other  solid  matter  is  such  that  the  ether  flows  freely  through  it, 
being  subjected  to  no  alteration  other  than  a  change  of  density. 
But  Stokes  has  shown  that  this  rather  startling  assumption  is 
probably  unnecessary,  if  the  speeds  of  the  earth  and  of  the  particles 
of  air  in  its  atmosphere  are  small  in  comparison  with  that  of  light. 


492  A    MANUAL    OF    PHYSICS. 

Of  course,  the  free  motion  of  the  earth  through  the  ether  shows 
that,  relatively  to  the  moving  earth,  the  ether  acts  like  a  practically 
non- viscous  fluid,  and  we  have  to  explain  how  it  is  that  the  ether 
can  act  both  like  a  fluid  and  like  an  elastic  solid.  No  explanation 
of  this  point  can  be  more  lucid  than  the  original  explanation  given 
by  Stokes  :  '  The  plasticity  of  lead  is  greater  than  that  of  iron  or 
copper,  and,  as  appears  from  experiment,  its  elasticity  is  less.  On  the 
whole  it  is  probable  that  the  greater  the  plasticity  of  a  substance, 
the  less  its  elasticity,  and  vice  versa,  although  this  rule  is  probably 
far  from  being  without  exception.  When  the  plasticity  of  the  sub- 
stance is  still  further  increased,  and  its  elasticity  diminished,  it 
passes  into  a  viscous  fluid.  There  seems  no  line  of  demarcation 
between  a  solid  and  a  viscous  fluid.  In  fact,  the  practical  dis- 
tinction between  these  two  classes  of  bodies  seems  to  depend  on  the 
intensity  of  the  extraneous  force  of  gravity,  compared  with  the 
intensity  of  the  forces  by  which  the  parts  of  the  substance  are  held 
together.  Thus,  what  on  the  Earth  is  a  soft  solid  might,  if  carried 
to  the  Sun,  and  retained  at  the  same  temperature,  be  a  viscous  fluid, 
the  force  of  gravity  at  the  surface  of  the  Sun  being  sufficient  to 
make  the  substance  spread  out  and  become  level  at  the  top ; 
while  what  on  the  Earth  is  a  viscous  fluid  might  on  the  surface  of 
Pallas  be  a  soft  solid.  The  gradation  of  viscous  into  what  are  called 
perfect  fluids  seems  to  present  as  little  abruptness  as  that  of  solids 
into  viscous  fluids  ;  and  some  experiments  which  have  been  made 
on  the  sudden  conversion  of  water  and  ether  into  vapour,  when  en- 
closed- in  strong  vessels  and  exposed  to  high  temperatures,  go 
towards  breaking  down  the  distinction  between  liquids  and  gases. 

'  According  to  the  law  of  continuity,  then,  we  should  expect  the 
property  of  elasticity  to  run  through  the  whole  series,  only  it  may 
become  insensible,  or  else  may  be  mastered,  by  some  other  more 
conspicuous  property.  It  must  be  remembered  that  the  elasticity 
here  spoken  of  is  that  which  consists  in  the  tangential  force  called 
into  action  by  a  displacement  of  continuous  sliding ;  the  displace- 
ments also  which  will  be  spoken  of  in  this  paragraph  must  be  under- 
stood to  be  such  displacements  as  are  independent  of  condensation 
or  rarefaction.  Now,  the  distinguishing  property  of  fluids  is  the 
extreme  mobility  of  their  parts.  According  to  the  views  explained 
in  this  article,  this  mobility  is  merely  an  extremely  great  plasticity, 
so  that  a  fluid  admits  of  a  finite,  but  exceedingly  small  amount  of 
constraint  before  it  will  be  relieved  from  its  state  of  tension  by  its 
molecules  assuming  new  positions  of  equilibrium.  Consequently 
the  same  oblique  pressures  can  be  called  into  action  in  a  fluid  as  in 
a  solid,  provided  the  amount  of  relative  displacement  of  the  parts  be 


THE    ETHEK.  493 

exceedingly  small.  All  we  know  for  certain  is  that  the  effect  of 
elasticity  in  fluids  [elasticity  of  form]  is  quite  insensible  in  cases  of 
equilibrium,  and  it  is  probably  insensible  in  all  ordinary  cases  of 
fluid  motion.  .  .  .  But  a  little  consideration  will  show  that  the 
property  of  elasticity  may  be  quite  insensible  in  ordinary  cases  of 
fluid  motion,  and  may  yet  be  that  on  which  the  phenomena  of  light 
entirely  depend.  When  we  find  a  vibrating  string,  the  small  extent 
of  vibration  of  which  can  be  actually  seen,  filling  a  whole  room  with 
sound,  and  remember  how  rapidly  the  intensity  of  the  vibrations  of 
the  air  must  diminish  as  the  distance  from  the  string  increases,  we 
may  easily  conceive  how  small  in  general  must  be  the  amount  of 
the  relative  motion  of  adjacent  particles  of  air  in  the  case  of  sound. 
Now,  the  extent  of  the  vibration  of  the  ether  in  the  case  of  light 
may  be  as  small,  compared  with  the  length  of  a  wave  of  light,  as 
that  of  the  air  is  compared  with  the  length  of  a  wave  of  sound  ;  we 
have  no  reason  to  suppose  it  otherwise.  When  we  remember,  then, 
that  the  length  of  a  wave  of  sound  in  air  varies  from  some  inches  to 
several  feet,  while  the  greatest  length  of  a  wave  of  light  is  about 
•00003  of  an  inch,  it  is  easy  to  imagine  that  the  relative  displace- 
ment of  the  particles  of  ether  may  be  so  small  as  not  to  reach,  nor 
even  come  near  to,  the  greatest  relative  displacement  which  could 
exist  without  the  molecules  of  the  medium  assuming  a  new  position 
of  equilibrium,  or,  to  keep  clear  of  the  idea  of  molecules,  without 
the  medium  assuming  a  new  arrangement  which  might  be  per- 
manent.' 

In  this  connection  Thomson  refers  to  the  properties  of  shoemakers' 
wax,  which  is  so  brittle  that  it  will  splinter  under  a  sudden  blow, 
and  which  will  flow  like  a  liquid  into  all  the  crevices  of  the  vessel 
which  contains  it,  while  leaden  bullets  will  sink  down  through  it, 
and  corks  will  float  up  through  it — if  only  sufficient  time  be  allowed 
(§  78).  The  resistance  to  the  passage  of  a  bullet  or  a  cork  through 
it  becomes  smaller  and  smaller,  the  slower  the  motion  becomes  ; 
and  it  may  be  that  the  motion  of  the  earth  through  the  ether  is  far 
less,  relatively  to  the  resisting  power  of  the  ether,  than  is  the  motion 
of  ..the  bullet  or  the  cork  relatively  to  the  resisting  power  of  the 
wax. 

380.  The  above  theory  of  the  ether  is  known  as  the  elastic-solid 
theory.  This  elastic  solid  cannot  possess  positive  compressibility, 
for  in  that  case  a  condensational-rarefactional  wave  —  of,  whose 
existence  we  have  no  experimental- evidence— might  be  propagated 
through  it  with  finite  speed.  Hence  Green,  who  investigated  the 
properties  of  this  ether,  assumed  that  it  was  incompressible.  He 
recognised  the  case  of  negative  compressibility,  but  dismissed  it  with 


494  A    MANUAL    OF    PHYSICS. 

the  assertion  that  a  medium  which  possesses  negative  compressi- 
bility— i.e.,  a  medium  which  expands  when  subjected  to  increased 
pressure,  and  contracts  when  pressure  is  removed — is  necessarily 
unstable. 

In  order,  on  this  theory,  to  account  for  the  reflection  and  refrac- 
tion of  light  at  an  interface,  Green  assumed  that  the  ether  had  the 
same  rigidity,  but  was  of  unequal  density,  on  the  two  sides  of  the 
interface.  This  gave  Fresnel's  law  in  the  case  of  vibrations  per- 
pendicular to  the  plane  of  reflection  ;  but,  in  the  case  of  vibrations 
in  the  plane  of  reflection,  it  gave  a  result  which  only  coincided 
with  Fresnel's  law  when  the  refractive  indices  of  the  two  media  were 
practically  identical. 

Sir  W.  Thomson  assumed  that  the  ether  consists  of  an  inviscid 
fluid  permeating  the  pores  of  an  incompressible  sponge -like  solid : 
but  the  result  deviated  further  from  Fresnel's  law  than  Green's  did. 
He  therefore  had  to  abandon  the  doctrine  of  incompressibility  ;  and, 
having  pointed  out  that  Green's  negatively  compressible  medium 
was  not  unstable  if  it  were  infinite  or  had  rigid  boundaries,  he 
assumed  such  negative  compressibility  as  to  make  the  velocity  of 
the  condensational-rarefactional  wave  zero.  He  assumed  (like 
Green)  equal  rigidities  of  the  ether  in  the  media  ;  but  this  condition 
has  been  shown  to  be  necessary  for  stability  when  the  other 
assumed  conditions  hold. 

This  contractile  ether  gives  FresnePs  laws ;  and  Glazebrook  has 
shown  that  it  explains  the  reflection  and  refraction  of  light  by 
transparent  bodies  and  by  metals,  double  refraction  and  dispersion 
(including  anomalous  dispersion),  and  that  it  gives  the  correct 
expression  for  the  velocity  of  light  in  a  moving  medium. 

Thomson  has  also  shown  how  a  model  of  a  medium  might  be 
constructed  by  rigid  jointed  connections  and  rigid  revolving  fly- 
wheels (or  gyrostats  in  which  a  frictionless  fluid  circulated  irrota- 
tionally)  which  has  no  intrinsic  rigidity,  i.e.,  no  intrinsic  elastic 
resistance  to  change  of  shape,  but  which  has  a  quasi-rigidity  due 
to  inherent  resistance  to  rotation ;  which  is  absolutely  devoid  of 
resistance  to  change  of  volume  or  to  irrotational  change  of  shape ; 
which  therefore  is  incapable  of  transmitting  condensational-rare- 
factional  waves,  but  which  can  transmit  vibrations  like  those  of 
light.  It  is  therefore  a  practical  realisation  of  his  contractile  ether. 

381.  The  electromagnetic  theory  of  the  ether  has  been  discussed 
in  the  last  chapter.  It  readily  explains  all  the  difficulties  which 
originally  beset  the  elastic -solid  theory ;  and  the  question  of  the 
condensational-rarefactional  wave  never  arises,  for  its  velocity  of 
propagation  is  infinite.  On  some  points  its  results  differ  from  those 


THE    ETHER.  495 

of  Thomson's  theory,  but  the  differences  are  too  small  to  admit  of 
crucial  tests  being  based  upon  them. 

It  must  be  observed,  however,  that  this  theory  stands  upon  a 
different  footing  from  the  former.  No  fundamental  assumption  is 
made  regarding  the  medium  other  than  that  it  shall  account  for 
certain  electrical  and  magnetic  actions. 

382.  The  property  of  dilatancy  in  a  medium  composed  of  rigid 
particles  in  contact  accounts  for  a  number  of  natural  phenomena 
and  presents  analogies  to  many  others. 

Let  us  suppose  that  we  have  a  space  filled  with  marbles  or  shot, 
each  being  in  contact  with  another  on  various  sides.  This  condi- 
tion can  be  satisfied  by  different  arrangements  of  the  spheres,  so 
that  in  some  arrangements  the  volume  occupied  by  a  given  number 
is  less  than  the  volume  occupied  by  the  same  number  in  other 
arrangements.  There  is  an  arrangement  of  maximum  volume  and 
an  arrangement  of  minimum  volume,  and  we  cannot  change  the 
hard  spheres  from  one  arrangement  to  another  without  altering 
the  volume.  (This  explains  the  meaning  of  the  term  '  dilatancy.') 
Change  of  shape  of  'such  a  mass  of  spheres  cannot  occur  without 
simultaneous  change  of  bulk.  Hence,  if  the  mass  be  enclosed  in 
an  inextensible,  but  flexible,  boundary,  no  change  of  shape  which 
necessitates  change  of  volume  can  occur. 

If  the  mass  of  spheres  is  enclosed  by  a  smooth  boundary,  motion 
of  the  layer  next  the  boundary  will  cause  less  alteration  of  volume 
than  does  the  motion  of  a  layer  in  the  interior  of  the  mass.  Hence, 
when  certain  stresses  are  applied,  there  may  be  a  streaming  motion 
of  the  spheres  along  the  boundary,  while  the  rest  of  the  spheres  do 
not  move.  This  conduction  of  the  parts  of  the  medium  along  a 
smooth  surface  resembles  the  conduction  of  electricity. 

If,  in  a  large  mass  of  spheres  in  the  condition  of  maximum 
density  enclosed  in  an  elastic  boundary,  one  sphere  grows  in  size, 
the  whole  medium  at  first  undergoes  dilatation.  Then  the  layer 
next  the  growing  sphere  reaches  the  condition  of  maximum  volume. 
After  that,  the  layer  next  the  sphere  will  be  returning  to  the 
condition  of  minimum  volume,  while  a  layer  a  little  farther  out  is 
at  maximum  volume.  Later,  there  will  be  a  succession  of  maxima 
and  minima  in  the  neighbourhood  of  the  growing  body. 

When  two  bodies,  growing  in  size,  are  present  in  the  medium  at 
a  considerable  distance  apart,  the  resultant  dilatation,  at  any  point, 
due  to  both  is  less  than  the  sum  of  the  separate  dilatations  at  that 
point  due  to  each.  Thus  there  will  be  a  force  of  attraction  between 
the  bodies  whose  magnitude  depends  upon  the  rate  at  which  the 
dilatation  varies  with  the  distance  between  the  bodies.  The  dilata- 


498  A   MANUAL    OF    PHYSICS. 

tion  becomes  periodic  when  the  bodies  are  near  to  each  other,  and 
attraction  and  repulsion  occur  alternately.  This  resembles  the 
phenomena  of  molecular  forces. 

Instead  of  supposing  that  the  boundary  is  elastic,  we  may  assume 
that  it  is  rigid,  and  that  the  growing  spheres  are  elastic.  We  may 
even  suppose  that  the  spheres  are  rigid  provided  that  the  medium 
is  composed  of  large  spheres  scattered  uniformly  among  small 
spheres,  for  such  a  medium  may  possess  elasticity  in  virtue  of  the 
propagation  of  distortional  waves  through  it — just  as  a  slack  chain 
possesses  elasticity  when  lateral  vibrations  are  passing  along  it. 
Even  if  the  small  grains  were  at  maximum  density,  distortional 
waves  could  pass,  the  distortions  of  the  two  sets  of  grains  being 
opposite.  This  may  throw  light  on  electrodynamic  and  magnetic 
phenomena.  Also,  the  separation  of  the  two  sets  of  grains  would 
produce  phenomena  analogous  to  those  depending  on  the  separation 
of  positive  and  negative  electricities.  And,  with  a  certain  arrange- 
ment of  the  large  and  small  grains,  the  state  of  stress  in  the  medium 
is  the  same  as  that  which  must  exist  in  the  ether  in  order  to 
account  for  gravity. 

Such,  in  brief  outline,  is  Osborne  Eeynolds'  theory  of  a  granular 
ether. 

383.  It  has  been  shown  also  that  an  ether  consisting  of  vortices  in 
a  perfect  fluid  might  be  capable  of  transmitting  light.     And  the 
instantaneous  propagation  of  gravitational  action  (if  it  be  instan- 
taneous) does  not  in  this  case  present  so  great  difficulty ;  for,  in  a 
certain  sense,   each  vortex  occupies  all  the    space  •v^iich  the  fluid 
fills — its  action  is  instantaneously  felt  in  all  parts. 

384.  It  is  not  to  be  supposed  that  all  these  theories  of  the  ether 
are  necessarily  antagonistic.     The  vortex  theory,  for  example,  may 
be  the  same  as  the  elastic-solid  theory. 


INDEX. 


THE   NUMBERS   REFER   TO   THE   SECTIONS. 


ABERRATION,  chromatic,  196 
spherical,  196 

Absorption,  co-efficient  of,  205 
Absorptive  power,  203,  256 
Acceleration,  42,  43,  44,  46,  50 
Accumulator,  water-dropping,  322 
Achromatism,  197 
Actinometer,  260 
Activity,  68 
Adiabatics,  295 
Amplitude,  51 
Analogy,  15 
Atmolysis,  112 
Atom,  82,  139-141 
Avogadro's  Law,  150 

BEATS,  musical,  175 

Biprism,  214 

Boiling,  275 

Bolometer,  34  3> 

Boyle's  Law,  103,  105,  149,  150 

BREWSTER,  203,  208,  217,  223,  238, 

243,  251,  263 
Brittleness,  83 

CALORIC,  254 
Capillarity,  119-127 
Carnot's  cycle,  290,  291 
CAUCHY,  146,  209,  258 
Caustics,  184,  189 
Cavendish  experiment,  89 
Characteristic  function,  201 
Charles'  Law,  150,  265 
Chemical  combination,  280 
Cohesion,  118,  137 
Colour,  180 

body,  205 

surface,  206 


Colours  of   crystalline  plates,  248, 

249 

mixed  plates.  222 
thick  plates,  223 
thin  plates,  218-221 
Compressibility,  79 

of  gases,  102 

of  a  perfect  gas,  104 

of  vapours,  106 

of  liquids,  113 

of  solids,  128 

Conductivity,  thermal,  151 
Conservation,  4,  12 

of  energy,  4,  10 

of  matter,  4,  5 
Consonance,  175 
Contact,  angle  of,  121 
Coronse,  232 

Coulomb's  torsion  balance,  312 
Couple,  70 

Critical  angle  of  reflection,  188 
Crucial  test,  15 
Crystalline  structure,  143,  144 
Curvature,  43 

DENSITY,  64 

of  earth,  89,  90 
Deviation,  iirioimum,  192 
Dew,  277 
Dialysis,  117 
Diamagnetism,  346 
Dichroism,  205 
Dielectrics,  306 
Diffraction  gratings,  233 
Diffusion  of  gases,  109,  151 

of  liquids,  116 
Diffusivity,  109 
Dilatancy,  382 

32 


498 


INDEX. 


Dilatation  of  gases,  265 

of  liquids,  264 

of  solids,  263 
Dimensions,  18,  67 
Discharging  rod,  314 
Dispersion,  irrationality  of,  197 

theories  of,  209 
Displacement,  40 
Disruptive  discharge,  321 
Dissociation,  152,  280 
Dissonanc^,  175,  176 
Divisibility,  79,  139 
Doppler's  principle,  178 
Dulong  and  Petit,  law  of,  270 
Dynamical    similarity,    67,    73,   76, 

'127,  168,  170,  171 
Dynamics,  63 
Dynamo,  367 

EFFICIENCY,  thermo-dynamic,  293 
Effusion,  110 
Elasticity,  79 

of  gases,  107 
of  liquids,  114 
of  solids,  134,  156 
.Electric  absorption,  321 
charge,  310 
current,  335,  343 
density,  316 
displacement,  319 
force,  lines  of,  318 
images,  317 
induction,  310,  319 
machines,  326 
polarisation,  338 
potential,  313 
quantity,  310 
v          resistance,  335,  343 
Electrodynamometer,  369 
Electrokinetic  energy,  368 
Electrolysis,  337 
Electromagnetic  induction,  367 
medium,  92 
units,  370 
waves,  376 
Electrometer,  325 
Electromotive  force,  313,  343 
Electrophorus,  311 
Electroscope,  gold-leaf,  309 
Electrostatic  capacity,  314 
energy,  320 
units,  370 

Emissivity,  203,  256 
Empirical  formulae,  17 
laws,  17 


Energy,  4,  7-11,  62 

conservation  of,  10 

dissipation  of,  11,  297,  298 

forms  of,  8 

kinetic,  8 

potential,  8,  1 1 

transformation  of,  9,  11 
Entropy,  296,  298 
Epoch,  51 

Equation,  personal,  16 
Equilibrium,  of  fluids,  75 

of  particles,  65 
of  rigid  bodies,  69 
stable,    15,    127,   331, 

357 

Equipotential  surfaces,  96 
Eriometer,  232 
Errors,  instrumental,  16 

observational,  16 

probable,  16 
Ether,  179,  202 
Evaporation,  152 
Exchanges,  theory  of,  255,  256 
Extension,  79 

FARADAY,  276,  315,  319,  337,  367, 

371 

Fluid,  equilibrium  of,  75 
motion  of,  61,  74 
motion  of  perfect,  74 
Fluorescence,  208,  253 
Focal  lines,  184,  189 
Focus,  principal,  183,  193 
Force,  62 

lines  and  tubes  of,  96 
moment  of,  70 
measurement  of,  64 
Forces,  composition  bf,  65 
molecular,  83,  125 
range  of,  145 
Form,  79 

Freedom,  degrees  of,  41,  54,  57 
Freezing  mixtures,  279 
FRESNEL,  186,  212,  214,  237,  239, 

240,  245 

FresnePd  rhomb,  252 
Friction,  62 

GALVANOMETER,  369 
Gas  battery,  341 
Gaseous  pressure,  149 
Geysers,  275 
Glaciers,  motion  of,  273 
Graphical  method,  17 
j   Gravitation,  8,  62,  85-101 


INDEX. 


499 


Gravitation,  law  of,  87 
Gravity,  centre  of,  88 

hypotheses  in  explanation 

of,  91,  92 
GREEN,  245,  380 
Gyration,  radius  of,  70 
Gyrostats,  58,  156 

HAIDINGER'S  brushes,  249 

Hall's  effects,  373 

Halos,  198 

HAMILTON,  Sir  W.  B.,  201,  246 

Harmonic  motion,  simple,  51,  52 

Harton  experiment,  90 

Heat,  conduction  of,  282,  287 

convection  of,  282,  288 

latent,  272,  274,  276 

mechanical  equivalent  of,  289 

radiant,  253 

radiation  of,  257-259,  261 

specific,  268,  269,  271,  301 

total,  276 
HELMHOLTZ,    Von,  173,  209,    249, 

338 

Hoarfrost,  277 
Hodograph,  48 
HOOKE,  135,  147,  237 
Hooke's  Law,  135 
Hope's  experiment,  264 
HUYGHENS,  186,  234,  235,  237 
Hygrometer,  277 
Hypothesis,  15 
Hysteresis,  355,  356,  357 

IMAGE,  182 

real,  183,  195 
•  virtual,  183,  195 
Impact,  68    , 
Impenetrability,  79 
Impulse,  68 

Indicator  diagram,  294-297 
Inductive  capacity,  specific,  315 
Inertia,  6,  79 

centre  of,  69,  88 
moment  of,  70,  71 
Isothermals,  24,  295 

JOULE,  147,  148,  289,  303,  357 
Joule's  Law,  342 

KEPLER'S  Laws,  86 
Kerr's  effects,  374 
Kinetics,  63 

KIRCHHOFP,  203,  204,  256 
Kirchhoff  8  Laws,  336 


LEAST  action,  200,  201 

confusion,  circle  of,  184 
time,  200,  201 . 

Lenses,  193,  194 

achromatic.  197 

LE  Boux,  207,  327,  333 

LE  SAGE,  91,  147 

Ley  den  jar,  314 

Light,  absorption  of,  202-209 
colour  of,  180 
diffraction  of,  224-233 
dispersion  of,  196,  197,  207, 

209 

intensity  of,  177 
interference  of,  210-223,  247 
nature  of,  179,  242 
polarisation  of,  237-252,  371 
propagation  of,  177,  186,  225 
radiation  of,  202-204 
reflection  of,  181-186,  240 
refraction  of,  187-200,  240 
refraction  (double),  234-236, 

244,  245,  250 
scattering  of,  181 
speed  of,  178 

theories    of,   179,    185,    186, 
199,  200,  375 

Lightning  rod,  322 

LLOYD,  213,  246 

MACCULLAGH,  239,  240,  243,  245 
Magnetic  axis,  349 

force,  lines  of,  350 

induction,  351 

intensity,  351 

moment,  349 

permeability,  352,  354 

pole,  349 

potential",  350 

retentiveness,  353 

shell,  362 

susceptibility,    352,    354, 
356,  357 

viscosity,  354,  355 
Magnetism,  residual,  353 
terrestrial,  359 
theories  of,  360,  365 
Magnetometer,  358 
Magnets,  coercive  force  in,  353,  356 

permanent,  344 

temporary,  344 
Malleability,  83 
MALUS,  201,  237,  238,  243    ' 
Mass,  7 

measurement  of,  64 


500 


INDEX. 


Matter,  4,  5 

definitions  of,  77 
general  properties  of,  79 
heterogeneity  of,  142,  145, 

146 

special  properties  of,  80,  83 
specific  properties  of,  81 
statts  of,  78 

MAXWELL,  24,  108,  145,  147,  152, 
153,  154,  250,  319,  320,  360,  372, 
375,  376 

Metallic  reflection,  206,  243 

Microscope,  195 

Mirage,  191 

Molecules,  82,  154 

size  of,  146 

Moment,  49 

Momentum,  64 

Motion,  laws  of,  63,  64,  65,  68 
of  centre  of  inertia,  69 
of  connected  particles,  68 
of  non-rigid  solids,  73 
of  particles,  64,  65,  66 
of  rigid  body,  56,  57 

Motor,  electric,  366 

Musical  intervals,  167 

NEBULAE  hypothesis,  93 
NEUMANN,  239,  240,  245 
NEWTON,  63,  67,  68,  87,  88,  92, 186, 

223,  237,  259 
Newton's  rings,  221 
Note,  157 

OHM'S  Law,  336,  338 
Osmose,  117 

PARAMAGNETISM,  346 
Paraselenes,  198 
Parhelia,  198 
Path,  mean  free,  148,  153 
Pendulum,  9 

ballistic,  68 
Peltier  effect,  331 
Period,  51,  53 
Phase,  51 

Phosphorescence,  208,  253 
Photometer,  177 
Piezometer,  113 
Polarising  prisms,  252 
Porosity,  79 
I'osition,  39 
Potential,  94 

gravitational,  95 
Precession,  58 


PREVOST,  147,  203,  255 
Primary  cells,  340 
Prisms,  192 
Projectiles,  47 
Pyrheliometer,  260 
Pyro-electricity,  232 
Pyrometers,  267 

QUARTZ  fibres,  89 

RADIOMETER,  153 
Rainbows,  198 
RANKIXE,  254,  295 
Reflection,  total,  188 
Refraction,  conical,  246 

index  of,  187,  223 
REGNAULT,  264,  265,  269,  271,  275, 

276,  277 

Residual  discharge,  321 
Resonance,  173 
Restitution,  co-efficient  of,  68 
Rigidity,  79,  83,  128,  131,  156 

flexural,  133 

torsional,  131 
Ripples,  76 
Rotation,  54,  ,55 
Rotational  equilibrium,  72 
Ruhmkorff  coil,  367 

SCALAR  quantity,  40 

Schehallion  experiment,  90 

Secondary  cells,  341 

Shear,  59 

Solar  radiation,  260 

Solenoid,  365 

Solution,  279 

Sound,  diffraction  of,  164 

intensity  of,  161 

interference  of,  165,  175 

pitch  of,  166 

propagation  of,  158 

quality  of,  174 

reflection  of,  162 

refraction  of,  163 

speed  of,  159,  160,  172 

wave-length  of,  157 
Spectrometer,  204 
Spectrum  analysis,  204 

diffraction,  2?3 
heat,  258 
refraction,  196 
Speed,  41 

average,  45 
mean  square,  148 
Specific  heat  of  electricity,  332 


INDEX. 


501 


Statics,  63 

STEWART,  BALFOUR,  203,  256,  259, 

261 
STOKES,  208,   217,    224,    225,   234, 

239,  242,  245,  379 
Strain,  59,  60 
Stress,  68 

Surface-tension,  120,  125,  126,  127 
Syren,  166 


TAIT,  153,  278,  284,  330,  333 
Telephone,  9 

Telescope,  astronomical,  195 
Temperature,  262 

absolute,  293 
absolute  zero  of,  266, 

303 

critical,  24,  278 
measurement  of,  267 
triple  point,  24 
Tenacity,  83,  138 
Tensor,  .42 
Theory,  15 

mathematical,  15 
Thermal  capacity,  268 

conductivity,  283,  284 
Therm< 'dynamic  motivity,  298 
Thermodynamics,  fi»st  law  of,  289 

second  law  of,  292 
Thermo-electric  diagram,  330,  333 

power,  328 
Thermometer,  267 

hypsometric,  275 
wet    and    dry   bulb, 

277 
Thermometric  conductivity,  284 

of  crystals,  286 
of  earth's  crust, 

285 
Thermopile,  328 


THOMSON,  15,  92,  101,  126,  137, 
141,  146,  148,  155,  209,  260,  273, 
292,  293 

Thomson  effect,  332 
Tone,  157 

combination,  176 
partial,  173 
Transpiration,  111 
Twist,  57 

VECTOR,  40 
Velocity,  41,  46 

angular,  43,  58 
average,  45 

Vibrations  of  air-columns,  172 
plates,  170 
rods,  168,  169 
strings,  17l_ 
Viscidity,  115 
Viscosity,  79,  83 

of  gases,  108,  151 
of  liquids,  115 
of  solids,  136 
Vortex  atoms,  141,  156 

motion,  61 
Vortices,  molecular,  254,  372 

WATER  equivalent,  269 

Wave-length,  53 
motion,  53 

Waves,  long  or  solitary,  76 

on  stretched  cords,  73 
oscillatory  or  free,  76 

Weight,  64,  79,  85 

Wheatstone's  bridge,  343 

Work,  7,  62 

YOUNG,  211,  221,  222,  232,  237 
Young's  modulus,  130 

ZONE  plates,  230 


THE   END. 


Bailliere,  Tindall  <k  Cox,  King  William  Street,  Strand. 


f 


Tb 


5  57")  98 


UNIVERSITY  OiF  CALIFORNIA  LIBRARY 


